Topological invariants from zero modes in systems of topological insulators (TI) and superconductors (TS). (16th August 2018) - Lin, Cheng-Hsiao; Tsai, Yan-Chr
Abstract: By a regularization method, a topological charge density formula expressed by the form of Dirac delta functions was derived. From this analytic approach, it will be easy to tell how topological charges are related to zero-field points (or zero modes) and depend on the differential properties of the associated vector field in the proximity of isolated zero modes. With this result, topological integrals (winding numbers) can be turned into simple sums contributed by isolated zero modes. Through this method, we investigated the physical features of zero modes for several cases including the integral form for the Teo-Kane model in dimensional reductions, the case of Majorana zero modes bound to a vortex line in a topological superconductor, the Bernevig-Hughes-Zhang (BHZ) and its extended (EBHZ) models, an extended Haldane model, and the additivity of topological charges. In these systems, the physical implications of zero modes which reflect their singular nature were also discussed. Without advanced knowledge, we provide a handy formula for calculating topological numbers accompanied by intuitive understandings of origins of topological charges which are identical to the zeros of the vector field.