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.


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.

| cite | improve this answer | |
  • $\begingroup$ I see. There is a problem with smoothness criterion. $\endgroup$ – user11937 Feb 22 at 7:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.