Crystalline symmetry in topological materials

Mathematics, Physics and Chemistry meet most harmonically through symmetry. Since Noether’s groundbreaking theorem, which established the connection between symmetry and a physical property, physicists have relied on symmetry arguments to understand the natural laws. On the other hand, the role of topology has been to some extent under-appreciated and linked to specific exotic phenomena. However, the two are not far apart: topology regards constants of motion of the space of states as a whole, reflected in nonlocal (as opposed to local) properties of physical systems, and it is reflected by anomalous behavior at spatial defects such as boundaries or vortices. In particular topology in band structure theory remained elusive for the largest portion of its history. Only the last decade has witnessed an enormous effort to bridge this gap, with remarkable success in the prediction of new physical phenomena. In this talk, I will focus on the interplay between topology and crystal symmetry in band structure theory. I will show how spatial (unitary) symmetries which were once expected to lead to trivial extensions of already known topological phenomena, can, in fact, introduce complexities that often defy our intuition but at the same time can be used to make topological phenomena more accessible to technological devices. I will show examples of how crystalline topological phases may cease to exist with periodic boundaries; how crystalline symmetry (and its breaking) may lead to the confinement of topological bands on crystalline defects such as stacking faults; and explored how the local symmetric environment may be manipulated through intrinsic mechanisms to construct surface-based topological devices such as a Majorana interferometer to probe non-abelian statistics.

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