Clarification on a statement Bernevig's textbook on Topological insulators

On page 11 of the aforementioned book Bernevig claims after doing some calculation that the integral of Berry curvature over a sphere containing a monopole is $$2\pi$$. Now my question is the following: the berry curvature is defined as the curl of the berry connection and hence its integral over a closed surface should vanish by the divergence theorem. How is this possible.

1 Answer

It hinges on the fact that the berry connection is not globally defined on the sphere. For Stokes' theorem(or Divergence theorem as you stated) to work, you need a vector field $${A}$$ defined on the whole manifold.

But if you choose a coordinate system on the sphere (e.g. spherical coordinate) and express the berry connection in that coordinate, you will find a point where the connection becomes singular. Even if you choose a different gauge to try removing the singularity, there is always one. See the second page of this lecture note for an example.

In sum, if the sphere in question encloses a monopole, the connection(hence curvature also) cannot be globally defined on the sphere, and this is the obstruction against application of Stokes' theorem.

• I see. There is a problem with smoothness criterion. – user11937 Feb 22 at 7:24