Sonic Landau Levels and Synthetic Gauge Fields in Mechanical Metamaterials

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).

Figure 1

(a) Mechanical graphene—a set of rods and nodes based on the honeycomb structure. The dashed line indicates the shape of a unit cell. (b) The lattice with a pure shear strain. (c) The shift of a Dirac point within the phonon spectrum of mechanical graphene due to the applied strain can be used to define an effective vector potential. (d) An externally applied nonuniform pure shear deformation that corresponds to a constant magnetic field. The external stress is applied by a torque $\tau$ on the boundary rods. (e) A nonuniform patterning of the local material stiffness that leads to a constant magnetic field. We consider periodic boundary conditions along $x$ and free boundary conditions along $y$.

Figure 2

Mechanical Landau levels: (a) A pseudomagnetic field leads to Landau levels around the Dirac point. (b) As the magnetic field increases, the zeroth-Landau-level band flattens. Band flatness can be characterized by the inverse magnetic length ${\ell }^{-1}$. (c) The inverse magnetic length scales as the square root of the magnetic field. (d) Density of states for the zeroth Landau level, for the same values of $B$ as in (b). The peak at the Dirac frequency rises as the bands flatten. (e) Visualizations of the zeroth Landau level at two different wave vectors. For $\mathbf{q}={\mathbf{q}}_K$, this mode has a Gaussian profile around the waveguide center, whereas far from this point, at $\mathbf{q}=0$, the mode decays exponentially away from the edge.

Figure 3

(a) Waveguide with two domain walls separating two regions with $\mu =0.08$ from a central region with $\mu =-0.08$. Bonds are colored according to their spring constants as in Fig. 1. Periodic boundary conditions are applied along $x$. (b) Variation of the effective Dirac mass $m(y)$ (dashed line) and midgap-mode amplitude for $q_x=2\pi /3a$ on either sublattice (solid lines). (c) Visualization of midgap mode with sublattices distinguished, showing strong sublattice polarization. Each point on the $A$ ($B$) sublattice is represented by a green (blue) disc whose area is proportional to the amplitude of the midgap mode. (d) Sublattice polarization of the domain-wall-bound mode in the presence of disorder in the spring constants [39]. (e) Even in the presence of strong (14%) disorder, we observe sublattice polarization due to the topological origin of the mode.

Figure 4

Single-mode response $\chi$ of Landau-level states in mechanical graphene, including the effect of damping on one sublattice and for pseudomagnetic field values (a) $B=0.0$ and (b) $B=0.3$. Colors correspond to the different Landau-level bands identified in Fig. 2. Insets: Wave number dependent attenuation rate $\eta$ of the corresponding bands. (c) The steady-state response (for $B=0.3$) to external periodic forcing with frequency close to the Dirac frequency and at an edge that is situated 50 unit cells to the left of the section shown. Each point is represented by a disc whose area is proportional to the amplitude of the response. (d) Zoom-in of (c) shows that the Landau-level mode is selectively enhanced due to the presence of sublattice-biased damping.