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

TL;DR

This work constructs a concrete microscopic 3d lattice realization of non-Abelian topological order by mapping the 3d quantum double model with group $G$ to the (3+1)D BF theory with an AAB twist, specifically identifying an isomorphism between the $\mathbb{D}_4$ lattice model and BF theory with $G=(\mathbb{Z}_2)^3$. It develops explicit lattice operators to create, fuse, and shrink particle and loop excitations, derives full fusion and shrinking data, and demonstrates that lattice shrinking rules are fully consistent with continuum fusion–shrinking relations. The study shows that non-Abelian shrinking channels can be controlled by internal loop degrees of freedom and verifies the fusion–shrinking consistency for Abelian and non-Abelian groups (including $\mathbb{D}_3$ and $\mathbb{D}_4$). By matching excitations and their algebraic data, the work provides a solid microscopic foundation for the Borromean-rings Borromean braiding scenario and bridges continuum topological field theory with exactly solvable lattice models, with implications for controllable higher-dimensional quantum matter and quantum simulation. It also outlines a pathway toward diagrammatic, higher-categorical descriptions and generalized symmetry structures in 3d topological orders, with potential extensions to fermionic systems and quantum computation in higher dimensions.

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.

Figures (22)

• Figure 1: Elementary fusion diagrams in 3d. $\mathsf{a}$, $\mathsf{b}$ and $\mathsf{c}$ denote excitations and we highlight them in blue. $\mu=\left\{1,2,\cdots,N_{\mathsf{c}}^{\mathsf{a}\mathsf{b}}\right\}$ labels different fusion channels with the same fusion output $\mathsf{c}$. Double-lines indicate that $\mathsf{a},\mathsf{b},\mathsf{c}\in \Phi _{0}^{3+1}$. If we want to draw a fusion diagram that represents fusion in the set $\Phi _{1}^{3+1}$, we only need to replace all the double-lines with single-lines.
• Figure 2: Shrinking diagram in 3d. The double-line and single-line represent excitations from $\Phi _{0}^{3+1}$ and $\Phi _{1}^{3+1}$ respectively. Since $\mathsf{a}$ and $\mathsf{b}$ are the input and output of the shrinking process respectively, $\mathsf{a}$ can be a loop or a particle while $\mathsf{b}$ can only be a particle. We use a triangle to represent the shrinking operation. This diagram can be defined as a vector and the orthogonal set $\left\{ \ket{\mathsf{a};\mathsf{b},\mu} ,\mu =1,2,\cdots ,S_{\mathsf{b}}^{\mathsf{a}} \right\}$ spans the shrinking space $V_{\mathsf{b}}^{\mathsf{a}}$ with $\mathrm{dim}\left( V_{\mathsf{b}}^{\mathsf{a}} \right) =S_{\mathsf{b}}^{\mathsf{a}}$.
• Figure 3: Fusing three excitations in 3d. $\mathsf{a}$, $\mathsf{b}$, and $\mathsf{c}$ are ultimately fuse into $\mathsf{d}$. The bracket "$\left(\mathsf{a},\mathsf{b}\right)$" is only used to emphasize that $\mathsf{a}$ and $\mathsf{b}$ fuse together first in the whole three-excitation fusion process and it does not change the vectors. The set of vectors $\left\{\ket{\left( \mathsf{a},\mathsf{b} \right) ;\mathsf{e},\mu} \otimes\ket{\mathsf{e},\mathsf{c};\mathsf{d},\nu}\right\}$ span the whole space $V_{\mathsf{d}}^{\mathsf{abc}}$, where different $\mu$, $\nu$ and $\mathsf{e}$ label different orthogonal vectors. The space $V_{\mathsf{d}}^{\mathsf{abc}}$ is isomorphic to $\oplus _{\mathsf{e}}(V_{\mathsf{e}}^{\mathsf{ab}}\otimes V_{\mathsf{d}}^{\mathsf{ec}})$. The dimension of $V_{\mathsf{d}}^{\mathsf{a}\mathsf{b}\mathsf{c}}$ is given by $\text{dim}(V_{\mathsf{d}}^{\mathsf{a}\mathsf{b}\mathsf{c}})=\sum_{\mathsf{e}}{N_{\mathsf{e}}^{\mathsf{ab}}}N_{\mathsf{d}}^{\mathsf{ec}}$. Here we only draw the diagram for fusion process in the set $\Phi _{0}^{3+1}$. One can obtain the diagram for subset $\Phi _{1}^{3+1}$ by simply replacing all double-lines with single-lines.
• Figure 4: Incorporating both fusion and shrinking processes in 3d. We stack two basic shrinking diagrams and a basic fusion diagram to describe the process $\mathcal{S} \left( \mathsf{a} \right) \otimes \mathcal{S} \left( \mathsf{b} \right)$.
• Figure 5: Local Hilbert space on a cubic lattice. Each edge is assigned with an arrow and a group element $g\in G$. Reversing the arrow of an edge changes the group element $g$ to its inverse $\bar{g}$.