Dirac Cones and Defect Charges

We consider the interplay between symmetry and topology in insulators and semi-metals. In particular, we have studied how crystalline symmetries affect the presence of band-crossing points in the bulk band structure, but also how the topology of rotation and inversion-symmetric insulators can be probed around topological defects, like edges and dislocations. We have shown that the crystalline topology can be directly related to edge states and excess charges. Moreover, we have shown that the presence of mirror and rotation symmetry can stabilize Dirac systems with up to twelfe inequivalent Dirac cones. Finally, we have shown how these insights can be used to engineer band structures.