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arXiv:2412.16681

Non-Invertible Symmetries as Condensation Defects in Finite-Group Gauge Theories

Clay Córdova, Davi Bastos Costa, Po-Shen Hsin

Published Dec 21, 2024• 2 citations
arXivHTMLPDF

Abstract

In recent work, we developed a method to construct invertible and non-invertible symmetries of finite-group gauge theories as topological domain walls on the lattice. In the present work, we consider abelian and non-abelian finite-group gauge theories in general spacetime dimension, and demonstrate how to realize these symmetries as condensation defects, i.e., as suitable insertions of lower dimensional topological operators. We then compute the fusion rules and action of these symmetries using their condensation expression and the algebraic properties of the lower dimensional objects that make them. We illustrate the discussion in $\mathbb{Z}_N$ gauge theory, where we derive the correspondence between domain walls, labeled by subgroups and actions for the doubled gauge group, and higher gauging condensation defects, labeled by subalgebras of the global symmetry. As a primary application, we obtain the condensation expression for the invertible symmetries of abelian gauge theories defined by outer automorphisms of the gauge group. We also show how to use these ideas to derive the action for certain non-abelian groups. For instance, one can obtain the action for the Dihedral group $\mathbb{D}_4$ by gauging a swap symmetry of $\mathbb{Z}_2\times\mathbb{Z}_2$ gauge theory.

DOI
10.48550/arxiv.2412.16681

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In recent work, we developed a method to construct invertible and non-invertible symmetries of finite-group gauge theories as topological domain walls on the lattice. In the present work, we consider abelian and non-abelian finite-group gauge theories in general spacetime dimension, and demonstrate how to realize these symmetries as condensation defects, i.e., as suitable insertions of lower dimensional topological operators. We then compute the fusion rules and action of these symmetries using their condensation expression and the algebraic properties of the lower dimensional objects that make them. We illustrate the discussion in ZN\mathbb{Z}_NZN​ gauge theory, where we derive the correspondence between domain walls, labeled by subgroups and actions for the doubled gauge group, and higher gauging condensation defects, labeled by subalgebras of the global symmetry. As a primary application, we obtain the condensation expression for the invertible symmetries of abelian gauge theories defined by outer automorphisms of the gauge group. We also show how to use these ideas to derive the action for certain non-abelian groups. For instance, one can obtain the action for the Dihedral group D4\mathbb{D}_4D4​ by gauging a swap symmetry of Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​ gauge theory.
Higher-form symmetries generalize conventional global symmetries and act on lower-dimensional submanifolds of a quantum system. While Abelian topological phases can be organized by 1-form symmetries that form a group, non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras. In this work, we make this statement precise in quantum double lattice models D(G)\mathcal D(G)D(G) for finite non-Abelian groups GGG. We construct the electric, magnetic, and dyonic 1-form operators directly at the lattice fixed point and show that together they form a complete nonlocal diagnostic algebra for the topological Hilbert space. Using these operators, we explicitly determine the cylinder and torus ground-state subspaces for arbitrary finite GGG. Furthermore, we calculate the microscopic fusion and gluing of the 1-form symmetries and show that their topological deformation properties emerge after projection to the defect-free topological subspace. Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking and provide an operator language that may be useful for characterizing such states in quantum processors.
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A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the d d -group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1) ( d + 1 ) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O_5] ∈ H^5(BG, U(1)) [ O 5 ] ∈ H 5 ( B G , U ( 1 ) ) obstruction that has appeared in prior work. We also show how the d d -group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the d d -group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2 ℤ 2 toric code.
We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category. We lay the foundations for a generalisation of this notion. By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations. For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations. Both this action and the monoidal structure are compatible with the hypergroup grading. The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad. Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.
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