Mechanical strain can lead to a synthetic gauge field that controls the dynamics of electrons in graphene sheets as well as light in photonic crystals. Here, we show how to engineer an analogous synthetic gauge field for lattice vibrations. Our approach relies on one of two strategies: shearing a honeycomb lattice of masses and springs or patterning its local material stiffness. As a result, vibrational spectra with discrete Landau levels are generated. Upon tuning the strength of the gauge field, we can control the density of states and transverse spatial confinement of sound in the metamaterial. We also show how this gauge field can be used to design waveguides in which sound propagates with robustness against disorder as a consequence of the change in topological polarization that occurs along a domain wall. By introducing dissipation, we can selectively enhance the domain-wall-bound topological sound mode, a feature that may potentially be exploited for the design of sound amplification by stimulated emission of radiation (SASER, the mechanical analogs of lasers). DOI: 10.1103/PhysRevLett.119.195502 Electronic systems subject to a uniform magnetic field experience a wealth of fascinating phenomena such as topological states [1] in the integer quantum Hall effect [2] and anyons associated with the fractional quantum Hall effect [3]. Recently, it has been shown that in a strained graphene sheet, electrons experience external potentials that can mimic the effects of a magnetic field, which results in the formation of Landau levels and edge states [4,5]. Working in direct analogy with this electronic setting, pseudomagnetic fields have been engineered by arranging CO molecules on a gold surface [6] and in photonic honeycomb-lattice metamaterials [7,8].In this Letter, we apply insights about wave propagation in the presence of a gauge field to acoustic phenomena in a nonuniform phononic crystal, using the appropriate mechanisms of strain-phonon coupling and frictional dissipation, in contrast to those present in electronic and photonic cases. The acoustic metamaterial context in which we implement gauge fields provides us with significant control [9-11] over frequency, wavelength, and attenuation scales unavailable in the analogous electronic realizations. For example, a metamaterial composed of stiff (e.g., metallic) components of micron-scale length may be suitable for control over ultrasound with gigahertz-scale frequencies, whereas cm-scale metamaterials may provide control over kHz-scale sound waves. We develop two strategies for realizing a uniform pseudomagnetic field in a metamaterial based on the honeycomb lattice, i.e., "mechanical graphene" [12]. In the first strategy, we apply stress at the boundary to obtain nonuniform strain in the bulk, which leads to a Landau-level spectrum, whereas in the second strategy, we exploit builtin, nonuniform patterning of the local metamaterial stiffness. This second strategy shows how the unique controllability of metamaterials can lead to novel designs inaccessibl...

Liquid crystals are complex fluids that allow exquisite control of light propagation thanks to their orientational order and optical anisotropy. Inspired by recent advances in liquid-crystal photo-patterning technology, we propose a soft-matter platform for assembling topological photonic materials that holds promise for protected unidirectional waveguides, sensors, and lasers. Crucial to our approach is to use spatial variations in the orientation of the nematic liquid-crystal molecules to emulate the time modulations needed in a so-called Floquet topological insulator. The varying orientation of the nematic director introduces a geometric phase that rotates the local optical axes. In conjunction with suitably designed structural properties, this geometric phase leads to the creation of topologically protected states of light. We propose and analyze in detail soft photonic realizations of two iconic topological systems: a Su–Schrieffer–Heeger chain and a Chern insulator. The use of soft building blocks potentially allows for reconfigurable systems that exploit the interplay between topological states of light and the underlying responsive medium.