Manipulation of Dirac Cones in Mechanical Graphene

Mechanical graphene, which is a spring-mass model with the honeycomb structure, is investigated. The vibration spectrum is dramatically changed by controlling only one parameter, spring tension at equilibrium. In the spectrum, there always exist Dirac cones at K- and K'-points. As the tension is modified, extra Dirac cones are created and annihilated in pairs. When the time reversal symmetry is broken by uniform rotation of the system, creation and annihilation of the Dirac cones result in a jump of the appropriately defined Chern number. Then, a flip of the propagation direction of the chiral edge modes takes place, which gives an experimental way to detect the topological transition. This is a bulk-edge correspondence of the classical system. We also demonstrate the other important concept, symmetry protection of the topological states, is at work in the classical system. For the time reversal invariant case, the topological edge modes exist for the fixed boundary condition but not for the free boundary condition. This contrast originates from the symmetry breaking at the free boundary.