Orthogonality relation for adiabatic Hamiltonian

Question

Suppose we have a Hamiltonian dependent on a parameter.
begin{equation}
hat{H}(lambda)=hat{H}_{0}+lambda hat{V}
end{equation}
We study how the energy eigenstates vary with the parameter by solving,
begin{equation}
hat{H}(lambda)|n,lambdarangle=E_{n}(lambda)|n,lambdarangle
end{equation}
The eigenstates are normalized for $langle m,lambda|n,lambdarangle=delta_{m,n}$. Appendix C
of the paper "BBGKY chain and kinetic equations for the level dynamics in an externally perturbed quantum system", https://arxiv.org/abs/1610.02380 claims that,
begin{equation}
langle m,lambda|frac{d}{dlambda}|m,lambdarangle=0
end{equation}
but I can only show that the real part of this complex number is zero. Could someone show me how to prove the imaginary part is also zero?

JoshuaTS · Answer

I didn't read the whole paper, but it appears that $lambda$ is a function of time, and that the Hamiltonian $H$ depends on time only through $lambda$. I say this because they appear to substitute $langle m,lambda|frac{partial}{partial lambda}|n,lambdarangledotlambda$ for $langle m,lambda|frac{partial}{partial t}|n,lambdarangle$ between equation 54 to equation 55, unless there's some non-trivial identity that is glossed over in that step. Under this assumption, the claim is not true. We can use the Schrödinger equation to prove otherwise:
$$E_m|m,lambdarangle=ifrac{partial}{partial t}|m,lambdarangle=ifrac{partial}{partial lambda}|m,lambdarangledot{lambda}$$
$$implies E_mlangle m,lambda|m,lambdarangle=ilangle m,lambda|frac{partial }{partiallambda}|m,lambdarangledotlambda.$$
Unless $E_m=0$, $langle m,lambda|frac{partial }{partiallambda}|m,lambdarangledotlambdaneq 0$.

Orthogonality relation for adiabatic Hamiltonian

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