Berry phase in degenerare case

  • Thread starter wdlang
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  • #1
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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 ?
 

Answers and Replies

  • #2
olgranpappy
Homework Helper
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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
[tex]
\frac{d}{dt} (e^{i\gamma(t)} n(t))=0
[/tex]
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
[tex]
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
[/tex]
so that (now using normalizaiton of n)
[tex]
i\frac{d\gamma}{dt} + \frac{d \vec\lambda}{dt}\cdot (n,\nabla n)=0
[/tex]

I guess, the Zee paper is just generalizing this to the case where instead of
[tex]
\eta_a = e^{i\gamma}\psi_a
[/tex]
with just a number for proportionality there is a matrix
[tex]
\eta_a=U_{ab}\psi_b\;.
[/tex]

I'm sure this isn't all too helpful, but maybe some other people will have more to say. cheers.
 
  • #3
307
0
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
[tex]
\frac{d}{dt} (e^{i\gamma(t)} n(t))=0
[/tex]
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
[tex]
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
[/tex]
so that (now using normalizaiton of n)
[tex]
i\frac{d\gamma}{dt} + \frac{d \vec\lambda}{dt}\cdot (n,\nabla n)=0
[/tex]

I guess, the Zee paper is just generalizing this to the case where instead of
[tex]
\eta_a = e^{i\gamma}\psi_a
[/tex]
with just a number for proportionality there is a matrix
[tex]
\eta_a=U_{ab}\psi_b\;.
[/tex]

I'm sure this isn't all too helpful, but maybe some other people will have more to say. cheers.
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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