# Berry's phase: in which space does the degeneracy appear?

This question follows a previous one of mine: Adiabatic theorem and Berry phase.

In his original paper [ M. V. Berry, Proc. R. Soc. Lond. A. Math. Phys. Sci. 392, 45 (1984) ], Berry discussed the possibility for an old-forgotten phase factor to arise after one loop is done in a parameter space, using the adiabatic theorem. This became famously known as a Berry's phase, see also Berry's curvature article on Wikipedia.

Soon after, [ Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987) ], AA proposed to generalised this adiabatic phase to non-adiabatic -- still cyclic -- evolutions.

What confused me is the sentence:

[..] we regard $\beta$ [the AA geometric phase] as a geometric phase associated with a closed curve in the projective Hilbert space and not the parameter space [...]

I would like to make this sentence a bit more clear, especially regarding the degeneracy point discussed in the previous question Adiabatic theorem and Berry phase.

What is the space in which the degeneracy appears?

Any comment to improve this question is welcome.

In Berry's paper, one considers a Hamiltonian depending on some external parameter $R$ (usually multidimensional); in the canonical example of a spin in an external magnetic field (Zeeman effect), the parameters are the 3 components of the magnetic field $\bf B$. For some values of the external parameter $R$, some eigenvalues may be degenerate. So degeneracies live in the abstract space of external parameter. For the spin example, the degeneracy appears at ${\bf B}=0$. The closed circuit discussed by Michael Berry takes place in the parameter space, and is driven externally and adiabatically.