Mechanical topological semimetals with massless quasiparticles and a finite Berry curvature

A topological quantum phase requires a finite momentum-space Berry curvature which, conventionally, can arise through breaking the inversion or the time-reversal symmetry so as to generate nontrivial, topologically invariant quantities associated with the underlying energy band structure (e.g., a finite Chern number). For conventional graphene or graphenelike two-dimensional (2D) systems with gapless Dirac cones, the symmetry breaking will make the system insulating due to lifting of the degeneracy. To design materials that simultaneously possess the two seemingly contradicting properties (i.e., a semimetal phase with gapless bulk Dirac-like cones and a finite Berry curvature) is of interest. We propose a 2D mechanical dice lattice system that exhibits precisely such properties. As a result, an intrinsic valley Hall effect can arise without compromising the carrier mobility as the quasiparticles remain massless. We also find that, with confinement along the zigzag edges, two distinct types of gapless edge states with opposite edge polarizations can arise, one with a finite but the other with zero group velocity.