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Berry phase in degenerare case

  1. Apr 29, 2008 #1
    Is there anyone familiar with berry phase?

    Wilczek and Zee have a classic paper PRL 52_2111

    I can not understand their equation (6)

    I can not see why the first equality should hold

    \eta_a and \eta_b should be orthogonal to each other, but why should \eta_b be orthogonal to the time derivative of \eta_a ?
  2. jcsd
  3. Apr 29, 2008 #2


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    hmmm... yeah. compare this to the usual berry's phase case where U_{ab} is just a number e^{i\gamma} (e.g., in Sakurai "Modern Quantum Mechanics" p. 465). in that case we have
    \frac{d}{dt} (e^{i\gamma(t)} n(t))=0
    where the time dependence in the n(t) (analogous to the \psi_a(t) in the Zee paper) is due to the changing parameter so that
    0=i\frac{d \gamma}{dt} e^{i\gamma} n + e^{i\gamma} \frac{d n}{dt}
    =i\frac{d\gamma}{dt} e^{i\gamma} n + e^{i\gamma} \frac{d \vec \lambda}{dt} \cdot \nabla_{(\lambda)} n
    so that (now using normalizaiton of n)
    i\frac{d\gamma}{dt} + \frac{d \vec\lambda}{dt}\cdot (n,\nabla n)=0

    I guess, the Zee paper is just generalizing this to the case where instead of
    \eta_a = e^{i\gamma}\psi_a
    with just a number for proportionality there is a matrix

    I'm sure this isn't all too helpful, but maybe some other people will have more to say. cheers.
  4. Apr 30, 2008 #3
    Thanks a lot!

    I read Simon's classic paper (PRL 51,2167), and i guess the equation in question reflects the parallel transport

    similar eqaution also appears in Simon's paper (on the second page, left upper part)
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