# Two Types of Non‐Abelian Topological Phase Transitions Under Duality Mapping in 1D Photonic Chains

## TL;DR

This paper discovers two new types of non-Abelian topological phase transitions in 1D photonic chains, revealing new mechanisms for topological phenomena.

## Contribution

The paper identifies two distinct non-Abelian phase transitions and their duality symmetry in photonic systems.

## Key findings

- Two types of non-Abelian phase transitions are revealed in photonic chains with p-orbital modes.
- Hidden duality symmetry separates distinct non-Abelian phases with identical Abelian invariants.
- Robust topological edge states are experimentally demonstrated in 1D photonic chains.

## Abstract

Exploring new topological phases and phenomena is essential in the modern physics. Although non‐Abelian topology with braided nodes in multiple bandgaps systems has been extensively explored, new non‐Abelian phase transitions with nodal line degeneracy still merit further theoretical classifications and experimental validations. Here, photonic chains with coupled p‐orbital modes are investigated where two types of non‐Abelian phase transitions are revealed. The first type is the braided‐node type, signified by the Dirac degeneracy node moving into or out of the unit circle. The second type corresponds to the sudden emerging of nodal line degeneracy which intersects with the unit circles. Hidden duality symmetry is unveiled in the rotation parameter space, in which two dual‐systems with identical Abelian topological invariants (specifically, Zak phase and Winding number) can be in distinct non‐Abelian topological phases, separated by the nodal‐line type phase transition. Rich non‐Abelian topological phases can be described by generalized quaternion group Q
16. Robust topological edge states are theoretically predicted by non‐Abelian bulk‐boundary correspondence and further experimentally demonstrated. This work expands new mechanism of non‐Abelian phase transition, paving the way to explore more topological phenomena and topological devices in various wave platforms.

In this work, two types of non‐Abelian phase transitions are revealed. The first type is the braided‐node type, signified by the Dirac degeneracy node moving into or out of the unit circle. The second type corresponds to the emerging of nodal‐line degeneracy which intersects with unit circles. Theoretical analyses and robust edge states, governed by non‐Abelian topological charge, are experimentally verified by 1D photonic chain.