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

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

2001-04-05

arxiv.org/abs/cond-mat/0104099v1

Physics

Condensed Matter

Strongly Correlated Electrons

4 pages and 1 jpeg figure

Scientific paper

10.1103/PhysRevA.64.052101

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.

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