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Berry's phase for compact Lie groups

Strahov, E

Published Nov 30, 2000• 16 citations
OpenAlex

Abstract

The Lie group adiabatic evolution determined by a Lie algebra parameter dependent Hamiltonian is considered. It is demonstrated that in the case when the parameter space of the Hamiltonian is a homogeneous K\\"ahler manifold its fundamental K\\"ahler potentials completely determine Berry geometrical phase factor. Explicit expressions for Berry vector potentials (Berry connections) and Berry curvatures are obtained using the complex parametrization of the Hamiltonian parameter space. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.

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arXiv:hep-th/9312173•Journal of Mathematical Physics•Mar 1, 1996•16 citations
The Lie group adiabatic evolution determined by a Lie algebra parameter dependent Hamiltonian is considered. It is demonstrated that in the case when the parameter space of the Hamiltonian is a homogeneous K\"ahler manifold its fundamental K\"ahler potentials completely determine Berry geometrical phase factor. Explicit expressions for Berry vector potentials (Berry connections) and Berry curvatures are obtained using the complex parametrization of the Hamiltonian parameter space. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.
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.
We study the total [dynamical plus geometrical (Berry)] phase of cyclic quantum motion for coherent states over homogeneous Kähler manifolds X=G/H, which can be considered as the phase spaces of classical systems and which are, in particular cases, coadjoint orbits of some Lie groups G. When the Hamiltonian is linear in the generators of a Lie group, both phases can be calculated exactly in terms of classical objects. In particular, the geometric phase is given by the symplectic area enclosed by the (purely classical) motion in the space of coherent states.
We study the total (dynamical plus geometrical (Berry)) phase of cyclic quantum motion for coherent states over homogeneous Kähler manifolds X = G/H, which can be considered as the phase spaces of classical systems and which are, in particular cases, coadjoint orbits of some Lie groups G. When the Hamiltonian is linear in the generators of a Lie group, both phases can be calculated exactly in terms of classical objects. In particular, the geometric phase is given by the symplectic area enclosed by the (purely classical) motion in the space of coherent states. I.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
The line bundles which arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the representation theory. The remarkable relation of the geometric phase to the classification of complex line bundles provides the necessary tools for establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry's connection. The results about the fibre bundles and group theory are used to introduce a procedure to reduce the problem of the non-adiabatic (geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-abelian monopoles is pointed out.
The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel–Weil–Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel–Weil–Bott theorem to Berry’s adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry’s connection. The results about the fiber bundles and group theory are used to introduce a procedure to reduce the problem of the nonadiabatic (geometric) phase to Berry’s adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-Abelian monopoles is pointed out.