Berry connection in SSH model

Question

In the SSH model for the 1D case, we get the eigenvectors as $$|(\pm)k>= \begin{pmatrix} \pm e^{-i\phi} \\ 1 \end{pmatrix}$$ where $\phi = tan^{-1}(\frac{wsin(k)}{v+wcos(k)})$. We can calculate the Berry connection as $i<k|\frac{d}{dk}|k>$, but I don't know how they get it equal to $-\frac{1}{2} \frac{d\phi}{dk}$. Can someone explain how do we arrive at this result, as I'm only getting it as $\frac{d \phi}{dk}$?

Answer 1

You must use normalized states in Berry's formula $i\langle k|\partial_k|k\rangle$, so you need to divide your $|k\rangle$ by $\sqrt 2$.