Berry’s phase for compact Lie groups

Abstract

The methods of Kähler geometry are applied to generalize the results of Berry obtained for SU(2) (namely, the existence of a geometrical part in the adiabatic phase) to any compact Lie group. We obtain explicit expressions for Berry’s geometric phases, Berry’s connections, and Berry’s curvatures in terms of parameters of the corresponding Lie algebra valued Hamiltonian. It is demonstrated that the parameter space of the Hamiltonian in the general theory is essentially a homogeneous Kähler manifold. The fundamental Kähler potentials of this manifold completely determine Berry’s phase. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.