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I have a question that has been bugging me recently. It's about the berry phase and something that I find contradictory.

One can see that it is possible to get rid of 2π x (integer) part of the Berry phase by means of a gauge transformation. This in general applies to phases (gauge transformation) that one can add to quantum states ##|ψ> → e^{\beta(R)} |ψ> ##, being R a bunch of parameters. Under a closed loop in R-space, single-valuedness of |ψ> demands ##\Delta \beta (R) = 2πn##, with n ∈ ℤ. So the Berry phase is defined up to shifts of ##2πn##.

Now, as far as I understand, in several places in literature, the same single-valuedness argument is used to impose that the Berry phase equals ##2πm## , with m ∈ ℤ !

For instance, in the Aharonov-Bohm example, I can define ##\lambda = -\phi/q ## , with ##\phi## being the azimuthal angle. Such a function is not globally defined, but ##A \rightarrow A + 1/q \nabla \phi## , ##|\psi> →| \psi> e^{i n \phi} ## transform in a single-valued way and hence ##\lambda## is an allowed gauge transformation. The effect of such ##\lambda## is piercing one extra unit of flux ##\Omega_0## (I can do the same process with an integer number of units of ##\Omega_0##). So the Berry phase, which in this case equals ##2\pi \Omega / \Omega_0## ( ##\Omega## is the total flux through the solenoid), is defined up to 2π×(integer). Only the non-integer part of ##\Omega /(2 \pi \Omega_0)## is gauge invariant.

So forthe particle going around a loop enclosing the flux and you get that the wavefunction is now ## e^{2 \pi i \Omega/ \Omega_0} |\psi>## and you can again worry about the single-valuedness of ##| \psi>##. And indeed, in some places they argue that ##\Omega## must be quantized in units of ##\Omega_0## to ensure single-valuedness. But, if ##\Omega / \Omega_0 \in \mathbb{Z}## I could gauge-away the Berry phase completely and in particular there would be no Aharonov-Bohm effect! This effect has been measured, so somehow ##\Omega / \Omega_0## can be chosen to be a non-integer.

I have seen similar arguments (using again single-valuedness) to show that the Berry phase under a periodic-in-time perturbation [Thouless, 1983] should be 2π×(integer). But then the same applies: couldn't I just gauge it away by a gauge tranformation ##|ψ> → e^{\beta(R)} |ψ> ## for a suitable ##\beta(R)##?

What am I missing?

Thank you very much and sorry for the long post!

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# A Berry phase and gauge dependence

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