# Adiabatic equation related to the Berry phase for lambda with first order terms

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: $$$$|\psi(t)\rangle=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}|n(t)\rangle \tag{1}$$$$

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: $$$$\phi(t)=\int_{\lambda(0)}^{\lambda(t)}A_n(\lambda)d\lambda \tag{2}$$$$ where $$A_n$$ is Berry connection.

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

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

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

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: $$$$(E_n-H_{\lambda})|\delta n\rangle=-i\hbar (\partial_{\lambda}+iA_n)|n\rangle. \tag{5}$$$$

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

• 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? – Nike Dattani Jul 29 '20 at 20:17
• 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. – jheindel Sep 1 '20 at 20:59
• @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! – Nike Dattani Dec 31 '20 at 16:53
• 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! – Nike Dattani Mar 19 at 15:56