Non-Abelian fusion rules from Abelian systems with SPT phases and graph topological order

TL;DR

The paper analyzes the Abelian class $H_{N} / \mathds{C}(\mathds{Z}_{P})$ of lattice gauge theories and proves that non-Abelian fusion rules arise and are necessary for SPT-phase transitions via condensation or symmetry breaking. By examining the $H_{3}/\mathds{C}(\mathds{Z}_{2})$ case and the structure of vertex/link operators $A_{v}$ and $C_{\ell}$ together with matter-excitation operators $W^{(J,K)}_{v}$, it generalizes non-Abelian fusion to broader $(N,P)$ families including cases with block-structured gauge actions and trivial representations. The work also connects two vacuum sectors in $H_{2P}/\mathds{C}(\mathds{Z}_{P})$ to $\mathds{Z}_{2P}\times\mathds{Z}_{2P}$ SPT phases and discusses phase transitions via symmetry breaking, using Dirac-sea-like analogies as a heuristic guide. Overall, the results extend the landscape of non-Abelian fusion beyond conventional quantum doubles and suggest directions for further exploration in related $D_{N}(\mathds{Z}_{P})$ theories and higher-dimensional systems.

Abstract

Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models, but does not prove whether these rules also exist in other cases, the purpose of this paper is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerate as a function of their algebra and, hence, they can support some symmetry-protected topological (SPT) phases, we prove that these non-Abelian fusion rules are always necessary for these SPT phase transitions to occur via a condensation mechanism or/and some global symmetry breaking.

Figures (2)

• Figure 1: Piece of an oriented square lattice $\mathcal{L} _{2}$ that supports the $H_{N} / \mathds{C} \left( \mathds{Z}_{P} \right)$ models, where we see the rose and light green coloured sectors respectively centred by the $v$-th vertex and $\ell$-th link of this lattice. Here, the highlighted links (in black) correspond to Hilbert subspaces in which, for instance, the vertex $A_{v}$ (the rose-coloured sector) and link $C_{\ell }$ (the light green coloured sector) operators act effectively. In the case of the links that structure the $v$-th vertex, they define a subset that we denote by $S_{v}$.
• Figure 2: Definition of the components $A^{g} _{v}$ and $C_{\ell , \alpha }$ in terms of their effective action on $\mathcal{L} _{2}$, where the group element $a$ is indexing an $\left\vert a \right\rangle$ basis element of the Hilbert space $\mathfrak{H} _{P}$ and the symbol $\alpha$ indexes an $\left\vert \alpha \right\rangle$ basis element of the Hilbert space $\mathfrak{H} _{N}$. Here, $\delta \left( x , y \right)$ should be interpreted as a Kronecker delta that was written differently for the sake of intelligibility (i.e., $\delta \left( x , y \right) = \delta _{xy}$).