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Dirac degeneracies are essential ingredients to control topological charge exchanges between bands and trigger the unique edge transport properties of topological materials. In addition, when Dirac cones are tilted, exotic phenomena can emerge such as anomalous Hall effect or unconventional Klein tunneling. However, the unique topological transport properties arising from the opening of tilted Dirac cone degeneracies have been left completely uncharted. Here, we demonstrate a new form of Dirac degeneracy that occurs in mechanical granular graphene (MGG): a tilted double Dirac cone, composed of two counter-tilted type-I Dirac cones. Different from the reported C6 systems, we show that the tilted double Dirac cone is present in a C2 granular graphene. Remarkably, a pair of anisotropic helical edge waves appears when the degeneracy is lifted. This leads to an anisotropic quantum spin-Hall topological insulator that possesses unique wave propagation properties, including anisotropic edge dispersion and direction-dependent edge-bulk mode conversion.

Tilted double Dirac cone and anisotropic quantum-spin-Hall topological insulator in mechanical granular graphene