We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

arXiv:1703.03418Physical review. B./Physical review. BJun 20, 201921 citations

Coupled-wire construction of non-Abelian higher-order topological phases

Jiaxin Pan, Longwen Zhou

Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.

cond-mat.mes-hallCross-listedarXiv:2512.21179Dec 25, 2025

Topological phases

Partha Ghose

Cambridge University Press eBooksApr 29, 19990 citations

Unsupervised learning of non-Abelian multi-gap topological phases

Xiangxu He, Ruo-Yang Zhang, Xiaohan Cui +2 more

Abstract Recent experiments have successfully realized multi-band non-Abelian topological insulators with parity-time symmetry. Their topological classification transcends the conventional ten-fold classification, necessitating the use of non-Abelian groups, manifesting novel properties that cannot be described using integer topological invariants. The unique non-commutative multiplication of non-Abelian groups, along with the distinct topological classifications in the context of homotopy with or without a fixed base point, makes the identification of different non-Abelian topological phases more nuanced and challenging than in the Abelian case. In this work, we present an unsupervised learning method based on diffusion maps to classify non-Abelian multi-gap topological phases. The automatically adiabatic pathfinding process in our method can correctly sort the samples in the same phase even though they are not connected by adiabatic paths in the sample set. Most importantly, our method can deduce the multiplication table of the non-Abelian topological charges in a data-driven manner without requiring a priori knowledge. Additionally, our algorithm can provide the correct classifications for the samples within both the homotopy with and without a fixed base point. Our results provide insights for future studies on non-Abelian phases using machine learning approaches.Abstract Recent experiments have successfully realized multi-band non-Abelian topological insulators with parity-time symmetry. Their topological classification transcends the conventional ten-fold classification, necessitating the use of non-Abelian groups, manifesting novel properties that cannot be described using integer topological invariants. The unique non-commutative multiplication of non-Abelian groups, along with the distinct topological classifications in the context of homotopy with or without a fixed base point, makes the identification of different non-Abelian topological phases more nuanced and challenging than in the Abelian case. In this work, we present an unsupervised learning method based on diffusion maps to classify non-Abelian multi-gap topological phases. The automatically adiabatic pathfinding process in our method can correctly sort the samples in the same phase even though they are not connected by adiabatic paths in the sample set. Most importantly, our method can deduce the multiplication table of the non-Abelian topological charges in a data-driven manner without requiring a priori knowledge. Additionally, our algorithm can provide the correct classifications for the samples within both the homotopy with and without a fixed base point. Our results provide insights for future studies on non-Abelian phases using machine learning approaches.

arXiv:2504.09198New Journal of PhysicsSep 1, 20250 citations

Geometric and topological phases

David S. Simon

Institute of Physics eBooksMay 1, 20210 citations

su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}

Moore-Read state, but the approach we outline here can be extended to general bosonic

su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}

topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.