Colloquium

Chern number analyis of a four-parameter Hamiltonian with $D_3$ symmetry

岩井 敏洋

4月15日(金) 13時30分

A discrete subgroup group $D_3$ of $SO(3)$ acts on the unit sphere, and also acts on $2\times 2$ Hermitian matices by adjoint action through a two-dimensional unitary representation. This manuscript deals with a four-parameter traceless Hermitian matrix (Hamiltonian) which is defined on the unit sphere and invariant under the $A_2\oplus E$ representation of $D_3$. If the eigenvalues of the Hamiltonian are not degenerate on the whole sphere, a complex line bundle is associated with the posiive (or negative) eigenvalue in such a manner that the eigenspace is assigned on each point of the sphere as a fiber. It is shown that the four-parameter space $\mathbb{R}2\times \mathbb{R}2$ is rduced to the two-torus $T2=S1\times S1$. On this torus, there exists a set of degenracy curves with each point of which eigenvalues are degenerate at some points of the sphere. The set of degeneracy curves divide the two-torus into several connected regions, on each of which a Chern number is assigned. Chern numbers are different from one another among regions.