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

We develop the topological band theory for systems described by non-Hermitian Hamiltonians, whose energy spectra are generally complex. After generalizing the notion of gapped band structures to the non-Hermitian case, we classify ``gapped'' bands in one and two dimensions by explicitly finding their topological invariants. We find nontrivial generalizations of the Chern number in two dimensions, and a new classification in one dimension, whose topology is determined by the energy dispersion rather than the energy eigenstates. We then study the bulk-edge correspondence and the topological phase transition in two dimensions. Different from the Hermitian case, the transition generically involves an extended intermediate phase with complex-energy band degeneracies at isolated ``exceptional points'' in momentum space. We also systematically classify all types of band degeneracies.

Topological Band Theory for Non-Hermitian Hamiltonians

We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.

Anomalous Edge State in a Non-Hermitian Lattice

Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of the Standard Model of particle physics. Their existence could explain the missing dark matter of the Universe. However, despite intensive searches, axions have yet to be observed. Here we show that magnetic fluctuations of topological insulators couple to the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. In particular, we show that the axion coupling enables a nonlinear modulation of the electromagnetic field, leading to attenuated total reflection. We propose a new optical-modulator device based on this principle. Axions are hypothetical particles that might play an important part in particle physics, astrophysics and cosmology. So far they have eluded observation, but theoretical work now predicts that axion physics might be explored in condensed-matter systems known as topological insulators.

http://absimage.aps.org/image/MAR10/MWS_MAR10-2009-001051.pdf

Dynamical Axion Field in Topological Magnetic Insulators

The extensive research of two-dimensional layered materials has revealed that valleys, as energy extrema in momentum space, could offer a new degree of freedom for carrying information. Based on this concept, researchers have predicted valley-Hall topological insulators that could support valley-polarized edge states at non-trivial domain walls. Recently, several kinds of photonic and sonic crystals have been proposed as classical counterparts of valley-Hall topological insulators. However, direct experimental observation of valley-polarized edge states in photonic crystals has remained difficult until now. Here, we demonstrate a designer surface plasmon crystal comprising metallic patterns deposited on a dielectric substrate, which can become a valley-Hall photonic topological insulator by exploiting the mirror-symmetry-breaking mechanism. Topological edge states with valley-dependent transport are directly visualized in the microwave regime. The observed edge states are confirmed to be fully valley-polarized through spatial Fourier transforms. Topological protection of the edge states at sharp corners is also experimentally demonstrated.

/pdf/direct-observation-of-valley-polarized-topological-edge-1hii122mqi.pdf

Direct observation of valley-polarized topological edge states in designer surface plasmon crystals.

The concept of gauge field is a cornerstone of modern physics and the synthetic gauge field has emerged as a new way to manipulate particles in many disciplines. In optics, several schemes of Abelian synthetic gauge fields have been proposed. Here, we introduce a new platform for realizing synthetic SU(2) non-Abelian gauge fields acting on two-dimensional optical waves in a wide class of anisotropic materials and discover novel phenomena. We show that a virtual non-Abelian Lorentz force arising from material anisotropy can induce light beams to travel along Zitterbewegung trajectories even in homogeneous media. We further design an optical non-Abelian Aharonov-Bohm system which results in the exotic spin density interference effect. We can extract the Wilson loop of an arbitrary closed optical path from a series of gauge fixed points in the interference fringes. Our scheme offers a new route to study SU(2) gauge field related physics using optics.

/pdf/non-abelian-gauge-field-optics-4smrelcb96.pdf

Non-Abelian gauge field optics.

Replication Data for: Experimental observation of non-Abelian topological charges and edge states

We have theoretically and experimentally achieved large-area one-way transport by using heterostructures consisting of a domain of an ordinary photonic crystal sandwiched between two domains of magnetic photonic crystals. The nonmagnetized domain carries two orthogonal one-way waveguide states which have amplitude uniformly distributed over a large area. We show that such one-way waveguide states can be used to abruptly narrow the beam width of an extended state to concentrate energy, and the transport is robust against different kinds of defects and imperfections. They are also immune to the Anderson-type localization when large randomness is introduced.

Topological One-Way Large-Area Waveguide States in Magnetic Photonic Crystals.

In this paper, we investigate the band properties of 2D honeycomb plasmonic lattices consisting of metallic nanoparticles. By means of the coupled dipole method and quasi-static approximation, we theoretically analyze the band structures stemming from near-field interaction of localized surface plasmon polaritons for both the infinite lattice and ribbons. Naturally, the interaction of point dipoles decouples into independent out-of-plane and in-plane polarizations. For the out-of-plane modes, both the bulk spectrum and the range of the momentum k ∥ where edge states exist in ribbons are similar to the electronic bands in graphene. Nevertheless, the in-plane polarized modes show significant differences, which do not only possess additional non-flat edge states in ribbons, but also have different distributions of the flat edge states in reciprocal space. For in-plane polarized modes, we derived the bulk-edge correspondence, namely, the relation between the number of flat edge states at a fixed , Zak phases of the bulk bands and the winding number associated with the bulk Hamiltonian, and verified it through four typical ribbon boundaries, i.e. zigzag, bearded zigzag, armchair, and bearded armchair. Our approach gives a new topological understanding of edge states in such plasmonic systems, and may also apply to other 2D 'vector wave' systems.

https://iopscience.iop.org/article/10.1088/1367-2630/18/10/103029/pdf

The Existence of Topological Edge States in Honeycomb Plasmonic Lattices

The topological invariants of a periodic system can be used to define the topological phase of each band and determine the existence of topological interface states within a certain bandgap. Here, we propose a scheme based on the full phase diagrams, and design the topological interface states within any specified bandgaps. As an example, here we propose a kind of one-dimensional phononic crystals. By connecting two semi-infinite structures with different topological phases, the interface states within any specific bandgap or their combinations can be achieved in a rational manner. The existence of interface states in a single bandgap, in all odd bandgaps, in all even bandgaps, or in all bandgaps, are verified in simulations and experiments. The scheme of full phase diagrams we introduce here can be extended to other kinds of periodic systems, such as photonic crystals and designer plasmonic crystals.

https://iopscience.iop.org/article/10.1088/1367-2630/aad136/pdf

Ruo-Yang Zhang

Papers

Non-Abelian gauge field optics.

Replication Data for: Experimental observation of non-Abelian topological charges and edge states

Topological One-Way Large-Area Waveguide States in Magnetic Photonic Crystals.

The Existence of Topological Edge States in Honeycomb Plasmonic Lattices

Designing topological interface states in phononic crystals based on the full phase diagrams