Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce that the sphere $S^2$ is the parameter space manifold. First question - shouldn't the parameter space actually be $S^2 \times (0,\infty) \cong B^2/O$, with $O$ being a ball center, as |B| $\in (0,\infty)$?

I would like to obtain more a priori understanding of the parameter space manifold identification. Working in Cartesian coordinates, for example, this $S^2$ identification is not transparent. My reasoning is along these lines - Fixing |B| one would naively first conclude that the $SO(3)$ is actually this manifold as we can orient B as we wish. However, we should ''subtract'' rotations around B and so we are lead to the coset manifold $SO(3)/SO(2)\cong S^2$. Letting |B| change, we finally arrive at $B^2/O$. Is this correct reasoning? Thing that puzzles me here is that we have spins here so $SU(2)$ should be more appropriate but how do we arrive to $S^2$ then?


The reason that the active parameter space is $S^2$ rather than the whole three dimensional space $\mathbb{R}^3$ or the ball $B^3$ is that the Berry curvature depends solely on the projector $P$ over the eigenspace, for example, an Abelian Berry curvature can be expressed as $$F = \mathrm{Tr}(P dP \wedge dP)$$ Now, spaces of projectors over one dimensional subspaces are projective spaces (which $S^2$ is a partiular case of): For two dimensional Hamiltonians, the projector on any one dimensional subspace can be casted in the form:

$$P(z, \bar{z}) = \frac{1}{1+\bar{z}z}\begin{bmatrix} 1 & z\\ \bar{z} & \bar{z}z \end{bmatrix}$$ with $$z = \tan \frac{\theta}{2}e^{i\phi}$$ and $\theta$ and $\phi$ are the coordinates parametrizing a unit sphere $S^2$. which is the one-(complex) dimensional projective space $\mathbb{CP}^1$.

This result generalizes as follows:

In fermionic time revesal invariant systems, the projectors are quaternionic valued and in the case of projectors on a one dimensional supspace (Abelian Berry phase), the parameter space is the one dimensional quaternionic projective space which is isomorphic to the four sphere:

$$\mathbb{HP}^1 \cong S^4$$

Please see for example: Avron, Sadun, Segert and Simon, where they obtain this parameter space as the parameter space of quadrupoles.

Another generalization is the case of the non-Abelian Wilczek-Zee phases, in this case the projectors are over higher dimensional subspaces of degenerate eigenvalues. Here, the parameter space is the Grassmann manifold:

$$\mathrm{Gr}(m,n) = U(m+n)/U(m)\times U(n)$$

where $m$ is the dimension of the Hamiltonian and $n$ is the dimension of the degenerate subspace of eigenvalues. Please see, for example, the article by Karle and Pachos.


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