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The Chern-Simons Invariant in the Berry Phase of a Two by Two Hamiltonian

The positive (negaive)-energy eigen vectors of the two by two Hamiltonian $H=ř\cdot\vec{\s}$ where $\vec{\s}$ are the Pauli matrices and $ř$ is a 3-vector, form a U(1) fiber bundle when $ř$ sweeps over a manifold $\cM$ in the three dimensional parameter space of $ř$ . For appropriately chosen base space $\cM$ the resulting fiber bundle can have non-trivial topology. For example when $\cM=S^2\equiv\{ř; |ř|=1\}$ the corresponding bundle has a non-zero Chern number, which is the indicator that it is topologically non-trivial. In this paper we construct a two by two Hamiltonian whose eigen bundle shows a more subtle topological non-triviality over $\cM=R^3\bigcup\{\infty\}$, the stereographic projection of $S^3$. This non-triviality is characterized by a non-zero Chern-Simons invariant.