Non-Abelian Berry’s phase, accidental degeneracy, and angular momentum

The non-Abelian Berry’s phase effect for a family of operators H0+k⋅V is considered, where H0 is rotationally invariant, V is a vector operator, and k varies over the unit vectors in R3. The parameter space is the two-sphere. The time evolution in the adiabatic limit is given by a connnection on a fiber bundle over the two-sphere. All connections consistent with the rotational symmetry are classified, and the time evolution is explicitly calculated for a nondegenerate Hamiltonian, as well as for a Hamiltonian with a double degeneracy. In the nondegenerate case, the connection is uniquely determined by the symmetry. In the doubly degenerate case, the connection is in some instances not determined by the symmetry. The case of approximate degeneracy is also discussed. A possible experimental test of this effect using optical pumping is described elsewhere.