## Abstract

BACKGROUND: Non-Abelian phenomena arise when the final state of a physical system is influenced by the sequence of operations. In such cases, we call these operations noncommutative, which stems from certain non-Abelian groups. Consider a book: If you rotate it 90° sequentially along two perpendicular axes, you will find that switching the orders of the two rotations will result in different final orientations of the book. This is known as the noncommutativity of the three-dimensional rotation of a classical rigid body. In modern physics, there are many examples in a similar vein. The angular momentum operators in quantum mechanics inherit the classical noncommutative nature, and the underlying non-Abelian group entails their quantization; non-Abelian gauge theory, a cornerstone of elementary particle physics, plays a central role in unifying electromagnetic forces, weak interactions, and quantum chromodynamics; and non-Abelian anyons, which obey non-Abelian braiding statistics, lie at the heart of the inception of fault-tolerant topological quantum computation. The shared salient feature in the examples above is that they all exhibit an expanded Hilbert space, which leads to matrix-valued operations that map to non-Abelian groups, that is, groups with elements that do not commute under multiplication. Because such operations can describe state evolution represented as matrices operating on multicomponent eigenvectors, the non-Abelian character of physical phenomena is manifested in both classical and quantum systems. Photonics and acoustics are excellent platforms to exemplify and explore non-Abelian phenomena owing to the large number of available degrees of freedom (such as electromagnetic duality, polarizations, angular momenta, and other synthetic dimensions) and the existing sophisticated techniques to control them. These fields have recently witnessed rapid developments in studying non-Abelian phenomena in synergy with quantum theories, condensed matter physics, and mathematical physics and complementarily introduce new perspectives on light and sound at the fundamental level.

ADVANCES: In this review, we discuss the theoretical foundations and experimental advances of non-Abelian physics in light and sound, epitomized by non-Abelian topological charges, non-Abelian gauge fields, non-Abelian mode dynamics, and non-Hermitian non-Abelian phenomena. In the well-established 10-fold–way classification, depending on how time-reversal, particle-hole, and chiral symmetries are satisfied, the topological phases of matter are categorized by Z or Z2 numbers, which are Abelian. Non-Abelian topological charges, beyond such a traditional Abelian framework, can be synthesized through transmission-line networks and metamaterials. In these cases, the underlying non-Abelian topological invariants are no longer integers but are represented by matrices like quaternions. They exhibit the non-Abelian features in multiple–energy band systems, enabling rich consequences such as path-dependent annihilation of Dirac points in two dimensions and the admissible link structure of nodal lines in three dimensions. Non-Abelian gauge fields are real-space manifestations of the non-Abelian Berry-Wilczek-Zee connection, which describes the adiabatic evolution of a set of multicomponent eigenstates and plays a crucial role in topological band theory. Recently, several schemes have emerged for the synthesis of these gauge fields; these include anisotropic metamaterials with tailored permittivity and permeability tensors under electromagnetic duality, multiple times of time-reversal symmetry breaking in different bases of Hilbert space at mode degeneracy in fibers and circuits, and exquisite control of synthetic spin-orbit interaction in exciton-polariton systems. Non-Abelian mode dynamics arise from the holonomic adiabatic evolution of multiple degenerate states. They differ from their Abelian counterparts because their resulting geometric phases also crucially depend on the ordering of the operations. Recent examples include the braiding, pumping, and Bloch oscillations of light and sound, ranging from microwave waveguides to integrated chips. Finally, non-Hermitian systems, that is, open systems that interact with environments, are receiving increasing attention: The complex nature of their eigenvalues and the nonorthogonality of their eigenvectors present a natural testbed for non-Abelian phenomena.

OUTLOOK: Judiciously achieving an expanded Hilbert space by internal degrees of freedom is a crucial prerequisite for studying non-Abelian physics in any system. To this end, opportunities are emerging in photonics and acoustics because of the rich numbers of available control knobs, such as duality, polarization, angular momentum, Bloch bands, particle numbers, symmetric-protected subspace, and gauge field–induced degeneracy. From an application point of view, non-Abelian gauge fields, braiding, and pumping could stimulate the development of mode-multiplexed devices and path-dependent topological mode converters. Except for the recent development of non-Hermitian braiding, most of the research results addressed here are within the linear Hermitian regime. We therefore anticipate that photonics and acoustics will foster richer non-Abelian physics in the nonlinear (i.e., effective interactions) and non-Hermitian regimes in the future.

ADVANCES: In this review, we discuss the theoretical foundations and experimental advances of non-Abelian physics in light and sound, epitomized by non-Abelian topological charges, non-Abelian gauge fields, non-Abelian mode dynamics, and non-Hermitian non-Abelian phenomena. In the well-established 10-fold–way classification, depending on how time-reversal, particle-hole, and chiral symmetries are satisfied, the topological phases of matter are categorized by Z or Z2 numbers, which are Abelian. Non-Abelian topological charges, beyond such a traditional Abelian framework, can be synthesized through transmission-line networks and metamaterials. In these cases, the underlying non-Abelian topological invariants are no longer integers but are represented by matrices like quaternions. They exhibit the non-Abelian features in multiple–energy band systems, enabling rich consequences such as path-dependent annihilation of Dirac points in two dimensions and the admissible link structure of nodal lines in three dimensions. Non-Abelian gauge fields are real-space manifestations of the non-Abelian Berry-Wilczek-Zee connection, which describes the adiabatic evolution of a set of multicomponent eigenstates and plays a crucial role in topological band theory. Recently, several schemes have emerged for the synthesis of these gauge fields; these include anisotropic metamaterials with tailored permittivity and permeability tensors under electromagnetic duality, multiple times of time-reversal symmetry breaking in different bases of Hilbert space at mode degeneracy in fibers and circuits, and exquisite control of synthetic spin-orbit interaction in exciton-polariton systems. Non-Abelian mode dynamics arise from the holonomic adiabatic evolution of multiple degenerate states. They differ from their Abelian counterparts because their resulting geometric phases also crucially depend on the ordering of the operations. Recent examples include the braiding, pumping, and Bloch oscillations of light and sound, ranging from microwave waveguides to integrated chips. Finally, non-Hermitian systems, that is, open systems that interact with environments, are receiving increasing attention: The complex nature of their eigenvalues and the nonorthogonality of their eigenvectors present a natural testbed for non-Abelian phenomena.

OUTLOOK: Judiciously achieving an expanded Hilbert space by internal degrees of freedom is a crucial prerequisite for studying non-Abelian physics in any system. To this end, opportunities are emerging in photonics and acoustics because of the rich numbers of available control knobs, such as duality, polarization, angular momentum, Bloch bands, particle numbers, symmetric-protected subspace, and gauge field–induced degeneracy. From an application point of view, non-Abelian gauge fields, braiding, and pumping could stimulate the development of mode-multiplexed devices and path-dependent topological mode converters. Except for the recent development of non-Hermitian braiding, most of the research results addressed here are within the linear Hermitian regime. We therefore anticipate that photonics and acoustics will foster richer non-Abelian physics in the nonlinear (i.e., effective interactions) and non-Hermitian regimes in the future.

Original language | English |
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Article number | eadf9621 |

Number of pages | 15 |

Journal | Science |

Volume | 383 |

Issue number | 6685 |

DOIs | |

Publication status | Published - 23 Feb 2024 |

## Scopus Subject Areas

- General