# Adiabatic theorem and Berry phase

As far as I can check, the adiabatic theorem in quantum mechanics can be proven exactly when there is no crossing between (pseudo-)time-evolved energy levels. To be a little bit more explicit, one describes a system using the Hamiltonian $H\left(s\right)$ verifying $H\left(s=0\right)=H_{0}$ and $H\left(s=1\right)=H_{1}$, with $s=\left(t_{1}-t_{0}\right)/T$, $t_{0,1}$ being the initial (final) time of the interaction switching. Then, at the time $t_{0}$, one has

$$H\left(0\right)=\sum_{i}\varepsilon_{i}\left(0\right)P_{i}\left(0\right)$$

with the $P_{i}$'s being the projectors to the eigenstates associated with the eigenvalue $\varepsilon_{i}\left(0\right)$, that we suppose known, i.e. $H_{0}$ can be exactly diagonalised. Then, the time evolution of the eigenstates is supposed to be given by

$$H\left(s\right)=\sum_{i}\varepsilon_{i}\left(s\right)P_{i}\left(s\right)$$

which is fairly good because it just requires that we are able to diagonalise the Hamiltonian at any time, what we can always do by Hermiticity criterion. The adiabatic theorem (see Messiah's book for instance)

$$\lim_{T\rightarrow\infty}U_{T}\left(s\right)P_{j}\left(0\right)=P_{j}\left(s\right)\lim_{T\rightarrow\infty}U_{T}\left(s\right)$$

with the operator $U_{T}\left(s\right)$ verifying the Schrödinger equation

$$\mathbf{i}\hslash\dfrac{\partial U_{T}}{\partial s}=TH\left(s\right)U_{T}\left(s\right)$$

can be proven exactly if $\varepsilon_{i}\left(s\right)\neq\varepsilon_{j}\left(s\right)$ at any time (see e.g. Messiah or Kato).

Now, the Berry phase is supposed to be non vanishingly small when we have a parametric curve winding close to a degeneracy, i.e. precisely when $\varepsilon_{i}\left(s\right) \approx \varepsilon_{j}\left(s\right)$. For more details, Berry defines the geometric phase as

$$\gamma_{n}\left(C\right)=-\iint_{C}d\mathbf{S}\cdot\mathbf{V}_{n}\left(\mathbf{R}\right)$$

with (I adapted the Berry's notation to mine)

$$\mathbf{V}_{n}\left(\mathbf{R}\right)=\Im\left\{ \sum_{m\neq n}\dfrac{\left\langle n\left(\mathbf{R}\right)\right|\nabla_{\mathbf{R}}H\left(\mathbf{R}\right)\left|m\left(\mathbf{R}\right)\right\rangle \times\left\langle m\left(\mathbf{R}\right)\right|\nabla_{\mathbf{R}}H\left(\mathbf{R}\right)\left|n\left(\mathbf{R}\right)\right\rangle }{\left(\varepsilon_{m}\left(\mathbf{R}\right)-\varepsilon_{n}\left(\mathbf{R}\right)\right)^{2}}\right\}$$

for a trajectory along the curve $C$ in the parameter space $\mathbf{R}\left(s\right)$. In particular, Berry defines the adiabatic evolution as following from the Hamiltonian $H\left(\mathbf{R}\left(s\right)\right)$, so a parametric evolution with respect to time $s$. These are eqs.(9) and (10) in the Berry's paper.

Later on (section 3), Berry argues that

The energy denominators in [the equation for $\mathbf{V}_{n}\left(\mathbf{R}\right)$ given above] show that if the circuit $C$ lies close to a point $\mathbf{R}^{\ast}$ in parameter space at which the state $n$ is involved in a degeneracy, then $\mathbf{V}_{n}\left(\mathbf{R}\right)$ and hence $\gamma_{n}\left(C\right)$, is dominated by the terms $m$ corresponding to the other states involved.

What annoys me is that the Berry phase argument uses explicitly the adiabatic theorem. So my question is desperately simple: what's the hell is going on there? Can we reconcile the adiabatic theorem with the Berry phase elaboration? Is the Berry phase a kind of correction (in a perturbative expansion sense) of the adiabatic theorem? Is there some criterion of proximity to the degeneracy that must be required in order to find the Berry phase?

REFERENCES:

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Could you please ask a more specific question than "what's going on"? What does it mean "What's going on"? You described what's going on. There are other things that are going on, too, but it is not clear which of them you find interesting or confusing. There's surely no contradiction in the text you wrote. Some theorems hold when the epsilons are safely separated, some effects appear when they're not, and so on. –  Luboš Motl Jul 11 '13 at 12:35
@LubošMotl Thanks for your comment, my question was written under rush. I've tried to add more focused questions, and more details. In short I want to know if the Berry phase and the adiabatic theorem are compatible, and to which extend they are (if they are). Please tell me if what I added is still insufficient to make any sense. Thanks again. –  FraSchelle Jul 11 '13 at 14:02
Berry phase and the adiabatic theorem are compatible. Some statements of the adiabatic theorem omit to mention the phase of the final wavefunction. Berry phase is an elaboration in that it says explicitly what that phase factor is. That's all there is to it. –  Dan Piponi Jul 11 '13 at 23:54
@DanPiponi Thanks for your comment. Would you then say that the proximity to a degeneracy point is not a problem at all, and that the adiabatic theorem can be expanded to include the degeneracy point(s) ? If yes, would you please elaborate a bit more about that. Thanks in advance. –  FraSchelle Jul 12 '13 at 7:52
@Oaoa: I wonder if part of the issue is the age of the references you are using? They were written before Berry's paper. They could be (implicitly) redefining the states to remove the Berry's phase, without considering the effect of closed loops in parameter space. Perhaps just a more modern reference like Nakahara would clear this up. Also proximity to degeneracy points is not an issue, if you go slowly enough, its going through degeneracy points that breaks the adiabatic theorem and hence berry's phase. –  BebopButUnsteady Jul 12 '13 at 14:19