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.

红宝石和星晶格的奇异电子能带结构以狄拉克锥和非平凡拓扑为特征，为二维 (2D) 狄拉克材料的研究提供了独特的平台。通常，理想的各向同性狄拉克锥受到时间反演对称性和反演的保护，因此其对晶格畸变的鲁棒性不仅具有根本意义，而且对实际应用也至关重要。在这里，我们系统地研究了红宝石晶格中狄拉克锥体的鲁棒性，以对抗四种典型的晶格畸变，这些畸变在从红宝石到星的过渡过程中破坏了反转和/或镜像对称。使用紧束缚方法，我们证明各向同性狄拉克锥及其相关拓扑特征在保持反演对称性的旋转扭曲晶格中保持完整（i-Ruby lattice）或面内镜像对称（m -Ruby lattice）。另一方面，狄拉克锥在a - 和b -Ruby 晶格中存在间隙，破坏了这些晶格对称性或反转。此外，识别旋转酉矩阵以将原始矩阵转换为扭曲的晶格。对称保护的狄拉克锥也在光子晶体系统中得到验证。非镜面对称i -Ruby 和非中心对称m -Ruby 晶格中显示的稳健狄拉克锥为二维狄拉克材料的设计提供了一般指导。