Suppose we have normalized states $| n(\vec{R})\rangle$ indexed by continuous variable $\vec{R}$. Then fixing our choice of gauge and ignoring dynamic phase, the phase difference between two states is the Berry's Phase:

$$\tag{1} \langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~=~ e^{i\gamma}$$

where, if $C$ is some curve that goes between $\vec{R}_0$ and $\vec{R}_0 + \Delta \vec{r}$,
$$\tag{2} \gamma~=~i\int_C \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R})\rangle \cdot d\vec{R}$$

If $\Delta \vec{r}$ is small, then
$$\tag{3} \gamma \approx i\langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r}$$

However, we can directly calculate this as well:

$$\langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~\approx~ \langle n(\vec{R}_0) | n(\vec{R}_0) \rangle + \langle n(\vec{R}_0) | \nabla_{\vec{R}}| n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r}$$

$$\tag{4} \approx 1 + \langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r} ~\approx~ \text{exp}\left[ \langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r} \right]$$

and therefore

$$\tag{5} \langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~\approx~ e^{-i\gamma} $$

There's a minus sign now! What am I doing wrong here?

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user ChickenGod