Non-Abelian geometric phase for general three-dimensional quantum systems

Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of CP2 is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians H = Q(i1...1n)J(i)j... J(in) and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian H = Q(ij)J(i)J(j) does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.