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In physics we often define the Chern number as the closed integral over the Berry curvature $$\Omega_{xy}=\frac{\partial A_x}{\partial k_y}-\frac{\partial A_x}{\partial k_x}.$$ With, $$A_i(\mathbf{k})=i\langle \psi_{\mathbf{k}}|\partial_{k_i}|\psi_{\mathbf{k}}\rangle.$$ So, we are often interested in, $$C=\frac{1}{2\pi}\oint_{}\Omega_{xy} d^2k,$$ where the Chern number is a closed integral over all momentum in the Brillouin zone.

My question is: What is a Chern number generally? Is it simply the integral over a closed surface such that the result is invariant in certain regions of the parameter space? I often see meantions of Chern classes, how are they related to Chern numbers?

Jens Roderus
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If M is an closed oriented $2n$-manifold, a Chern number of $M$ is simply the integral of any product of Chern classes of its tangent bundle which has total degree $2n$.

Mariano Suárez-Álvarez
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  • Could you elaborate on Chern classes and tangent bundles for someone who is not familiar with those kinds of things? Or point me to some nice source for this information. – Jens Roderus Feb 19 '18 at 20:29
  • Tangent bundles are explained in every textbook on manifolds. Lee's book, for example, is particularly great. As for Chern classes, I suggest Husemoller's book and/or the one by Stasheff (but if you do not knw what the tangent bundle is I doubt either could be of much use, really) – Mariano Suárez-Álvarez Feb 19 '18 at 22:41
  • One more question. If a Chern number is an integral of any product of Chern classes of a manifold's tangent bundle, then, is the thing that appears in my integral such a product? Is the Berry curvature a product of Chern classes? – Jens Roderus Feb 20 '18 at 06:59
  • It is a product of 1 class. – Mariano Suárez-Álvarez Feb 20 '18 at 16:58
  • Cool. I think your information is helpfull. Thank you very much. – Jens Roderus Feb 20 '18 at 18:45