Inversion/Mirror Symmetry-Protected Dirac Cones in Distorted Ruby Lattices

The exotic electronic band structures of Ruby and Star lattices, characterized by Dirac cone and nontrivial topology, offer a unique platform for the study of two-dimensional (2D) Dirac materials. In general, an ideal isotropic Dirac cone is protected by time reversal symmetry and inversion, so that its robustness against lattice distortion is not only of fundamental interest but also crucial to practical applications. Here we systematically investigate the robustness of Dirac cone in a Ruby lattice against four typical lattice distortions that break the inversion and/or mirror symmetry in the transition from Ruby to Star. Using a tight-binding approach, we show that the isotropic Dirac cones and their related topological features remain intact in the rotationally distorted lattices that preserve the inversion symmetry (i-Ruby lattice) or the in-plane mirror symmetry (m-Ruby lattice). On the other hand, the Dirac cones are gapped in the a- and b-Ruby lattices that break both these lattice symmetries or inversion. Furthermore, a rotational unitary matrix is identified to transform the original into the distorted lattice. The symmetry-protected Dirac cones were also verified in photonic crystal systems. The robust Dirac cones revealed in the non-mirror symmetric i-Ruby and non-centrosymmetric m-Ruby lattices provide a general guidance for the design of 2D Dirac materials.