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$$\mathcal{A}_{k}(\lambda) \equiv -i \langle n|\frac{\partial}{\partial \lambda^{k}}|n\rangle,$$

where the ket ##|n(\lambda)\rangle## depends on the parameters ##\lambda^{k}, k=1,2,\dots## in the system.

The field strength ##\mathcal{F}_{kl}## of the Berry connection ##\mathcal{A}_{k}(\lambda)## is defined by

$$\mathcal{F}_{kl} = \frac{\partial\mathcal{k}}{\partial\lambda_{l}}-\frac{\partial\mathcal{l}}{\partial\lambda_{k}}.$$

Therefore, we can define an analog of Maxwell's theory with the Berry connection ##\mathcal{A}_{k}(\lambda)##. As such, we expect the Berry connection ##\mathcal{A}_{k}(\lambda)## to be gauge invariant. In other words, there must be a gauge redundancy in the definition of the Berry connection ##\mathcal{A}_{k}(\lambda)##

I was wondering if you guys have any idea about the physical meaning of this gauge redundancy for some state ##|n(\lambda)\rangle##.