Show that under a local gauge transformation (|n, boldsymbol{R}angle ightarrow e^{i chi(boldsymbol{R})}|n, boldsymbol{R}angle), the Berry connection (boldsymbol{A}_{n}(boldsymbol{R})) is not invariant, transforming instead as in

Show that under a local gauge transformation \(|n, \boldsymbol{R}angle \rightarrow e^{i \chi(\boldsymbol{R})}|n, \boldsymbol{R}angle\), the Berry connection \(\boldsymbol{A}_{n}(\boldsymbol{R})\) is not invariant, transforming instead as in Eq. (27.21).

Data from Eq. 27.21

An(R) An(R)- VRX (R)