Chern-Simons invariant in the Berry phase of a $2\times 2$ Hamiltonian

By varying $\mathrm{}(x,y,z)\mathrm{}$ within a manifold $\mathcal{M},$ the positive (negaive)-energy eigenvectors of the $2\times 2$ Hamiltonian $H=x{\sigma }_x+y{\sigma }_y+z{\sigma }_z$ (where ${\sigma }_{x,y,z}$ are the Pauli matrices) form a U(1) fiber bundle. For certain $\mathcal{M}$ the bundle has nontrivial topology. For example, when $\mathcal{M}{=S}^2$ the associated bundle has nonzero Chern number indicating that it is topologically nontrivial at the highest level. In this paper we construct a simple $2\times 2$ Hamiltonian whose eigenvector bundle exhibits a more subtle topological nontriviality when $\mathcal{M}$ is a closed three manifold. This nontrivial topology is characterized by nonzero Chern-Simons invariant.