Mechanical Graphene

Abstract

We present a model of a mechanical system with a vibrational mode spectrum identical to the spectrum of electronic excitations in a tight-binding model of graphene. The model consists of point masses connected by elastic couplings, called "tri-bonds," that implement certain three-body interactions, which can be tuned by varying parameters that correspond to the relative hopping amplitudes on the different bond directions in graphene. In the mechanical model, this is accomplished by varying the location of a pivot point that determines the allowed rigid rotations of a single tri-bond. The infinite system constitutes a Maxwell lattice, with the number of degrees of freedom equal to the number of constraints imposed by the tri-bonds. We construct the equilibrium and compatibility matrices and analyze the model's phase diagram, which includes spectra with Weyl points for some placements of the pivot and topologically polarized phases for others. We then discuss the edge modes and associated states of self stress for strips cut from the periodic lattice. Finally, we suggest a physical realization of the tri-bond, which allows access to parameter regimes not available to experiments on (strained) graphene and may be used to create other two-dimensional mechanical metamaterials with different spectral features.

10 pages, 4 figures; Submitted to New Journal of Physics focus issue on Topological Mechanics

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• Directional mechanical response in the bulk of topological metamaterials
• Universal Continuum Theory for Topological Edge Soft Modes
• Phonon rotoelectric effect
• Topological Mechanics from Supersymmetry
• Edge states of mechanical diamond and its topological origin
• Weyl points of mechanical diamond
• Omnimodal topological polarization of bilayer networks: analysis in the Maxwell limit and experiments on a 3D-printed prototype
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Soft topological modes protected by symmetry in rigid mechanical metamaterials
• Topological mechanical metamaterial with nonrectilinear constraints
• First Order Topological Phase Transitions and Disorder Induced Majorana Modes in Interacting Fermion Chains
• Zero-frequency corner modes in mechanical graphene
• SAW interdigitated transducers as topological mechanical metamaterial