Non-abelian Topological Phases Achieve Multigap Protection with a Second-Order Insulator

Beyond conventional understandings of materials’ topological properties, researchers are now exploring states of matter defined by non-Abelian topological charges, which promise new avenues for advanced physics and potential technological applications. Jiaxin Pan and Longwen Zhou, both from Ocean University of China, alongside their colleagues, demonstrate a novel method for constructing these complex states within higher-order topological insulators. The team proposes a coupled-wire construction that allows for the creation of non-Abelian higher-order topological phases, revealing a system where corner modes are protected by fundamental symmetries and described by a unique combination of non-Abelian and Abelian topological characteristics. This achievement extends the known boundaries of higher-order topology and provides a pathway towards realising these exotic states in practical, synthetic materials like those used in photonics and acoustics.

The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries, and described by a topological vector. This vector combines a non-Abelian quaternion charge with an Abelian winding number, offering a novel characterisation of these states. Corner states appear only when both the quaternion charge and winding number are nontrivial, while weak topological edge states emerge when only the quaternion charge is nontrivial, expanding the understanding of how bulk properties relate to edge and corner behaviour.

Topological Phases in Driven Photonic Crystals

Current research extensively investigates topological insulators, particularly higher-order versions, and their realisation in diverse physical systems. A significant focus lies on non-Abelian topological phases, theoretically predicted and experimentally observed in photonic, phononic, and other platforms. The use of time-periodic driving, known as Floquet topology, is also a very active area, enabling the creation and manipulation of topological states. Photonic and phononic crystals, along with metamaterials, serve as common platforms for experimental studies of these phenomena. Investigations into non-Hermitian physics are also expanding the understanding of topological phases, with particular attention paid to corner and edge states, the defining features of higher-order and conventional topological insulators, respectively.

Experimental observations are crucial, and research demonstrates these effects in systems like phononics, photonics, ultracold atoms, and potentially 2D materials. Foundational work on topological band structures, such as that by Asbóth and Zak, underpins these investigations, while studies by Benalcazar, Bernevig, and Hughes have established the basis for higher-order topology. The field is also exploring non-abelian topological phases, which can host exotic states with potential applications in topological quantum computation, with researchers focusing on characterising these phases through frame rotation perspectives.

Hybridizing Abelian and Non-Abelian Topologies Unveiled

This research introduces a new theoretical framework for constructing both intrinsic and hybridized non-Abelian higher-order topological insulators, extending the understanding of these materials. Scientists demonstrate that the topology of these insulators arises from the combination of topological charges originating from both Abelian and non-Abelian mathematical groups within the material’s structure, revealing a rich landscape of topological phases in two-dimensional lattices. The team identified these phases using a two-component topological charge, linking a non-Abelian quaternion with a conventional integer winding number, and established that nontrivial values for both components are necessary for the emergence of protected corner states. The investigation further reveals a complete correspondence between the bulk topological properties of the material and the behavior of its edge and corner states, solidifying a unified understanding of bulk-edge-corner correspondence in non-Abelian systems. Researchers discovered weak topological edge states with non-Abelian origins, potentially offering pathways for applications in quantum information and computation due to their unique braiding properties. While the current work focuses on specific lattice models, exploring more complex systems and investigating the effects of disorder represent important avenues for future research, with transmission line networks offering a promising platform for experimental realisation and probing of these theoretical predictions.