Non-invertible Lattice 1-Form Symmetries for Non-Abelian Topological Order

arXiv:2608.16520v1 Announce Type: cross Abstract: 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 $\mathcal D(G)$ for finite non-Abelian groups $G$. 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 $G$. 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.

Abstract: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 $\mathcal D(G)$ for finite non-Abelian groups $G$. 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 $G$. 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.

Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Lattice (hep-lat); Quantum Physics (quant-ph)

arXiv:2608.16520 [cond-mat.str-el]

(or arXiv:2608.16520v1 [cond-mat.str-el] for this version)

https://doi.org/10.48550/arXiv.2608.16520 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Non-invertible Lattice 1-Form Symmetries for Non-Abelian Topological Order

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