The positive (negaive)-energy eigen vectors of the two by two Hamiltonian $H=\v{ r}\cdot\vec{ \s}$ where $\vec{ \s}$ are the Pauli matrices and $\v{ r}$ is a 3-vector, form a U(1) fiber bundle when $\v{ r}$ sweeps over a manifold $\cM$ in the three dimensional parameter space of $\v{ r}$ . For appropriately chosen base space $\cM$ the resulting fiber bundle can have non-trivial topology. For example when $\cM=S^2\equiv\{ \v{ r}; |\v{ r}|=1\}$ 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.
Categories: cond-mat.str-el cond-mat.stat-mech