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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.