Manipulation of Dirac Cones in Mechanical Graphene

Abstract

Recently, quantum Hall state analogs in classical mechanics attract much attention from topological points of view. Topology is not only for mathematicians but also quite useful in a quantum world. Further it even governs the Newton’s law of motion. One of the advantages of classical systems over solid state materials is its clear controllability. Here we investigate mechanical graphene, which is a spring-mass model with the honeycomb structure as a typical mechanical model with nontrivial topological phenomena. The vibration spectrum of mechanical graphene is characterized by Dirac cones serving as sources of topological nontriviality. We find that the spectrum has dramatic dependence on the spring tension at equilibrium as a natural control parameter, i.e., creation and annihilation of the Dirac particles are realized as the tension increases. Just by rotating the system, the manipulated Dirac particles lead to topological transition, i.e., a jump of the “Chern number” occurs associated with flipping of propagating direction of chiral edge modes. This is a bulk-edge correspondence governed by the Newton’s law. A simple observation that in-gap edge modes exist only at the fixed boundary, but not at the free one, is attributed to the symmetry protection of topological phases.

Model for topological phononics and phonon diode

“…As learned from electrons, the QH and QAH effects are characterized by a topological invariant called Chern number of the first class, which is defined as an integration of the Berry curvature over the Brillouin zone and can be nonzero only for broken TRS [12]. Nontrivial topological states with nonzero Chern number have been well established for photons [13][14][15] and recently reported for phonons, acoustic or mechanical systems [16][17][18][19][20][21][22][23][24][25][26][27]. As demonstrated by these previous works, topological quantities (like Berry curvature, Berry phase and Chern number) are independent of particle statistics and can be defined for phonons as for electrons.…”

“…As demonstrated by these previous works, topological quantities (like Berry curvature, Berry phase and Chern number) are independent of particle statistics and can be defined for phonons as for electrons. The phononic states with nonzero Chern number give the Q(A)H-like edge states of phonons, which are topologically protected to be gapless, one-way and immune to backscattering [16][17][18][19][20][21][22][23][24][25][26][27].…”

“…Theoretically, a few works proposed to break TRS of phonons by spin-lattice interactions in magnetic materials, Coriolis/magnetic field and optomechanical interactions [17,19,20,22,[28][29][30]. But a unified theoretical description of all these interactions is still lacking.…”

Sonic Landau Levels and Synthetic Gauge Fields in Mechanical Metamaterials

“…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.…”

“…1(a)] [12]. The compressional stiffness of the rods κ is determined by their fixed Young's modulus E, variable cross-section area S, and length a via ES=a.…”