Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

Higher-order topological insulators1–5 are a family of recently predicted topological phases of matter that obey an extended topological bulk–boundary correspondence principle. For example, a two-dimensional (2D) second-order topological insulator does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D topological insulator, but instead has topologically protected zero-dimensional (0D) corner states. The first prediction of a second-order topological insulator1, based on quantized quadrupole polarization, was demonstrated in classical mechanical6 and electromagnetic7,8 metamaterials. Here we experimentally realize a second-order topological insulator in an acoustic metamaterial, based on a ‘breathing’ kagome lattice9 that has zero quadrupole polarization but a non-trivial bulk topology characterized by quantized Wannier centres2,9,10. Unlike previous higher-order topological insulator realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically protected but reconfigurable local resonances. A second-order topological insulator in an acoustical metamaterial with a breathing kagome lattice, supporting one-dimensional edge states and zero-dimensional corner states is demonstrated.

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Acoustic higher-order topological insulator on a kagome lattice.

Recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi-)edge modes. We then apply the K-theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z2 topological index for arbitrary quantum channels. Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.

Topological phases of non-Hermitian systems

Topological Phases of Non-Hermitian Systems

The study of classical wave physics has been reinvigorated by incorporating the concept of the geometric phase, which has its roots in optics, and topological notions that were previously explored in condensed matter physics. Recently, sound waves and a variety of mechanical systems have emerged as excellent platforms that exemplify the universality and diversity of topological phases. In this Review, we introduce the essential physical concepts that underpin various classes of topological phenomena realized in acoustic and mechanical systems: Dirac points, the quantum Hall, quantum spin Hall and valley Hall effects, Floquet topological phases, 3D gapless states and Weyl crystals. This Review describes topological phenomena that can be realized in acoustic and mechanical systems. Methods of symmetry breaking are described, along with the consequences and rich phenomena that emerge.

Topological phases in acoustic and mechanical systems

The acoustic analogue of a topological insulator is shown: a metamaterial exhibiting one-way sound transport along its edge. The system — a graphene-like array of stainless-steel rods — is a promising new platform for exploring topological phenomena.

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Acoustic topological insulator and robust one-way sound transport

Acoustics is a classical field of study that has witnessed tremendous developments over the past 25 years. Driven by the novel acoustic effects underpinned by phononic crystals with periodic modulation of elastic building blocks in wavelength scale and acoustic metamaterials with localized resonant units in subwavelength scale, researchers in diverse disciplines of physics, mathematics, and engineering have pushed the boundary of possibilities beyond those long held as unbreakable limits. More recently, structure designs guided by the physics of graphene and topological electronic states of matter have further broadened the whole field of acoustic metamaterials by phenomena that reproduce the quantum effects classically. Use of active energy-gain components, directed by the parity-time reversal symmetry principle, has led to some previously unexpected wave characteristics. It is the intention of this review to trace historically these exciting developments, substantiated by brief accounts of the salient milestones. The latter can include, but are not limited to, zero/negative refraction, subwavelength imaging, sound cloaking, total sound absorption, metasurface and phase engineering, Dirac physics and topology-inspired acoustic engineering, non-Hermitian parity-time synthetic active metamaterials, and one-way propagation of sound waves. These developments may underpin the next generation of acoustic materials and devices, and offer new methods for sound manipulation, leading to exciting applications in noise reduction, imaging, sensing and navigation, as well as communications.

Breaking the barriers: advances in acoustic functional materials

This article reports on the experimental realization of a flow-free, pseudospin-based acoustic topological insulator designed using the strategy of zone folding. Robust sound one-way propagation is demonstrated with the presence of non-spin-mixing defects. On the other hand, it is shown that spin-mixing defects, which break the geometric symmetry and therefore the pseudo-time-reversal symmetry, can open up nontrivial band gaps within the edge state frequency band, and their width can be tailored by the extent of the defect. This provides a possible route for realizing tunable acoustic topological insulators.

Observation of zone folding induced acoustic topological insulators and the role of spin-mixing defects

Topologically nontrivial states in acoustic systems, characterized by one-way sound transport that is immune to backscattering, have been explored in recent years. This study experimentally demonstrates the acoustic Weyl points and surface Fermi arcs in a $t\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}e\ensuremath{-}d\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l$ phononic crystal. The topological robustness of the surface states is confirmed, and surface dispersions are engineered by tuning the boundary conditions. These findings pave the way for the further development of topological acoustics in three dimensions, and can be used for acoustic spatial filtering, wave guiding, and harvesting of sound energy.

Experimental Observation of Acoustic Weyl Points and Topological Surface States

Topological valley states at the domain wall between two artificial crystals with opposite valley Chern numbers offer a feasible way to realize robust wave transport since only broken spatial symmetry is required. In addition to the valley, spin and crystal dimension are two other important degrees of freedom, particularly in realizing spin-related topological phenomena. Here we experimentally demonstrate that it is possible to construct two-dimensional acoustic topological pseudospin-valley coupled saddle surface states, designed from glide symmetry in a three-dimensional system. By taking advantage of such two-dimensional surface states, a full set of acoustic pseudospins can be realized, exhibiting pseudospin-valley dependent transport. Furthermore, due to the hyperbolic character of the dispersion of saddle surface states, multi-directional anisotropic controllable robust sound transport with little backscattering is observed. Our findings may open research frontiers for acoustic pseudospins and provide a satisfactory platform for exploring unique acoustic topological properties in three-dimensional structures.

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Hao Ge

Papers

Acoustic topological insulator and robust one-way sound transport

Breaking the barriers: advances in acoustic functional materials

Observation of zone folding induced acoustic topological insulators and the role of spin-mixing defects

Experimental Observation of Acoustic Weyl Points and Topological Surface States

Three-dimensional topological acoustic crystals with pseudospin-valley coupled saddle surface states.