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

• URL: http://arxiv.org/abs/2608.16520v1
• Date: Mon, 17 Aug 2026 13:03:36 GMT
• Title: Non-invertible Lattice 1-Form Symmetries for Non-Abelian Topological Order
• Abstract summary: We show that non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras.<n>Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking.
• Score: 0.007496778384043148
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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