Topological Insulators and Superconductors a transport view

 In  order to compute physical properties in a crystal we need to transport spinors in the Brilouine Zone.

This is done with the help of the connection and curvature of the crystal space.Using symmetries such as Time rversal ,Inversion and Charge conjugation we can decide if the system is topological or trivial.

Insulators with an inverted band are topological.The boundary of such a crystal is metallic and chiral.

In the presence of disorder the "connections" give rise to a \pi Berry phase.

As a result the system remains metallic in two dimensions.

In the presence of an attractive interaction one obtains superconductivity. Due to the \pi Berry phase the vortices are half vortices named Majorana Fermions.

We analyze the finger print of such a Topological Superconductor. We couple metallic leads and compute the Andreev Crossed reflection (which is different from the regular Andreev reflection).

We compute different cases a)Single vortex b) two vortices

c) Coupled mettalic rings pierced by flux( Persistent -current).