Consider the following derivation in David Vanderbilt's book "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" (2018, Cambridge University Press (page 100).

The wave function for the adiabatic approach is as follows: \begin{equation} |\psi(t)\rangle=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}|n(t)\rangle \tag{1} \end{equation}

where $e^{i\phi(\lambda(t))}$ is the geometric phase or Berry phase and $e^{-i\gamma(t)}$ is dynamic phase.

Berry phase has the following mathematical expression: \begin{equation} \phi(t)=\int_{\lambda(0)}^{\lambda(t)}A_n(\lambda)d\lambda \tag{2} \end{equation} where $A_n$ is Berry connection.

Dynamic phase has the following mathematical expression: \begin{equation} \gamma(t)=\frac{1}{\hbar}\int_{0}^{t}E_n(t')dt' \tag{3} \end{equation}

Expanding equation number (1) with first order $\lambda$ terms:

\begin{equation} |\psi(t)\rangle=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}\big[|n(\lambda)\rangle+\dot{\lambda}|\delta n(t)\rangle\big] \tag{4} \end{equation}

Equation (4) solves the time-dependent Schrödinger equation to order zero in $\dot{\lambda}$, but we now require that it should also do so at first order.

For this purpose we can discard terms that scale like $\ddot{\lambda}$ or $\lambda^2$. You get: \begin{equation} (E_n-H_{\lambda})|\delta n\rangle=-i\hbar (\partial_{\lambda}+iA_n)|n\rangle. \tag{5} \end{equation}

How did they get equation (5) from equation (4)?

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    $\begingroup$ I don't know what all the symbols mean here, and don't have that book, so can't look up what they mean easily. But Eq. 6 seems to be somewhat common: I've been working today on describing p-DMRG for my types of DMRG question, and you can see that Eq. 9 in this: arxiv.org/abs/1803.07150 is very similar to Eq. 6 in your question. Eq. 5 is similar. Maybe we just need to know what all these symbols mean? $\endgroup$ – Nike Dattani Jul 29 '20 at 20:17
  • $\begingroup$ Eq. 5 is a direct application of the expression for first-order correction from perturbation theory. I don't totally understand eq. 6, but it is somehow leveraging the completeness of the basis. In particular, I just don't know what the notation $|\partial_{\lambda}n\rangle$ means. $\endgroup$ – jheindel Sep 1 '20 at 20:59
  • $\begingroup$ @jheindel: We have a policy not to ask 2 questions in one post, and combined with the fact that the notation was not made clear, the question was closed. However, I have deleted the question about Eq. 6, so if you can write an answer about Eq. 5 only, it would be very appreciated as it would help clean up our unanswered queue for the upcoming new year! $\endgroup$ – Nike Dattani Dec 31 '20 at 16:53
  • $\begingroup$ Hi @jheindel ! Since this is now one of our longest standing unanswered questions, and has been left up for over 6 months, I wonder if you might be able to spare a couple minutes to answer the part about Eq. 5? The part about Eq. 6 was deleted. I'm trying to clear up the unanswered queue! $\endgroup$ – Nike Dattani Mar 19 at 15:56

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