On symmetrization of 6j-symbols and Levin-Wen Hamiltonian

TL;DR

This work analyzes when $6j$-symbols in spherical fusion categories can enjoy full tetrahedral symmetry (symmetrized $6j$-symbols) and demonstrates that such symmetry is not universal by presenting category $\mathcal{E}$ as a counterexample. It introduces mirror conjugate symmetry for unitary spherical categories and shows that an appropriate normalization yields this symmetry, enabling unitary, well-behaved $6j$-symbols even when symmetrization fails. The authors extend the Levin-Wen exact-solvability framework to unitary spherical categories endowed with mirror-conjugate symmetric $6j$-symbols, proving that the honeycomb lattice Hamiltonian is exactly soluble and Hermitian in this setting. The paper also provides explicit normalization and $F$-matrix data for category $\mathcal{E}$, illustrating both the impossibility of global symmetrization and the viability of the proposed normalization. Overall, it deepens the connection between spherical fusion categories, topological lattice models, and exactly solvable Hamiltonians, with implications for related topological quantum field theories.

Abstract

It is known that every ribbon category with unimodality allows symmetrized $6j$-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category $\mathcal{E}$. We define the mirror conjugate symmetry of $6j$-symbols instead and show that $6j$-symbols of any unitary spherical category can be normalized to have this property. As an application, we discuss an exactly soluble model on a honeycomb lattice. We prove that the Levin-Wen Hamiltonian is exactly soluble and hermitian on a unitary spherical category.

Figures (1)

• Figure 1: Honeycomb lattice model and operator $B^s_P$

Theorems & Definitions (9)

• Definition 2.0.1
• Definition 2.0.2
• Theorem 3.3.1
• proof
• Corollary 3.3.2
• proof
• Corollary 3.3.3
• proof
• Theorem 5.0.1