Bridging Microscopic Constructions and Continuum Topological Field Theory of Three-Dimensional Non-Abelian Topological Order

Abstract: Here we provide a microscopic lattice construction of excitations, fusion, and shrinking in a non-Abelian topological order by studying the three-dimensional quantum double model. We explicitly construct lattice operators that create, fuse, and shrink particle and loop excitations, systematically derive their fusion and shrinking rules, and demonstrate that non-Abelian shrinking channels can be controllably selected through internal degrees of freedom of loop operators. Most importantly, we show that the lattice shrinking rules obey the fusion--shrinking consistency relations predicted by twisted $BF$ field theory, providing solid evidence for the validity of field-theoretical principles developed over the past years. In particular, we compute the full set of excitations, fusion, and shrinking data at the microscopic lattice level and verify exact agreement between the microscopic $\mathbb{D}_4$ quantum double lattice model and the continuum $BF$ field theory with an $AAB$ twist and $(\mathbb{Z}_2)^3$ gauge group, thereby placing the latter field theory, originally discovered in 2018 in connection with Borromean-ring braiding, on a solid microscopic footing. Our results bridge continuum topological field theory and exactly solvable lattice models, elevate fusion--shrinking consistency from a continuum field-theoretical principle to a genuine topological phenomenon defined at the microscopic lattice scale, and provide a concrete microscopic foundation for experimentally engineering higher-dimensional non-Abelian topological orders in controllable quantum simulators, such as trapped-ion systems.