 01145 WuYi Hsiang and DungHai Lee
 The ChernSimons Invariant in the Berry Phase of a Two by Two Hamiltonian
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Apr 18, 01

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Abstract. 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 3vector, 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 nontrivial topology. For example when $\cM=S^2\equiv\{\v r; \v r=1\}$ the corresponding bundle
has a nonzero Chern number, which is the indicator that it is topologically nontrivial. In this paper we
construct a two by two Hamiltonian whose eigen bundle shows a more subtle topological nontriviality over
$\cM=R^3\bigcup\{\infty\}$, the stereographic projection of $S^3$. This nontriviality is characterized by a
nonzero ChernSimons invariant.
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