Show that the Berry curvature (27.24) can also be written as

\[\Omega_{\mu v}^{n}(\boldsymbol{R})=i\left(\left\langle\partial_{\mu} n(\boldsymbol{R}) \mid \partial_{v} n(\boldsymbol{R})\rightangle-\left\langle\partial_{v} n(\boldsymbol{R}) \mid \partial_{\mu} n(\boldsymbol{R})\rightangle\right),\]

where \(\partial_{\alpha} \equiv \partial / \partial R_{\alpha}\) and the definition (27.16) of the Berry connection \(A_{\mu}^{n}\) was used.

Data from Eq. 27.16

Data from Eq. 27.24

Transcribed Image Text:

Vn(R) = n(R) An(R) = ih (n, RVR|n, R),