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Berry's phase for compact Lie groups

Strahov, E

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 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.

Nov 30, 200016 citations

Correspondence between Berry's phase and Lewis's phase for quadratic Hamiltonians

Daniel A. Morales

The authors show the existence of a relation between Berry's phase and Lewis's phase for the case of quadratic Hamiltonians.

Journal of Physics A Mathematical and GeneralSep 21, 198868 citations

Berry phase in homogeneous Kähler manifolds with linear Hamiltonians

Luis J. Boya, A. M. Perelomov, Mariano Sant

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.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.

Sep 14, 201313 citations

BERRY PHASE IN YANG–BAXTER SYSTEMS AND BOGOLIUBOV HAMILTONIAN AS A DERIVATIVE OF DIRAC HAMILTONIAN VIA BRAID RELATION

Mo‐Lin Ge, Kang Xue, Jing‐Ling Chen

Conference in Honor of C N Yang's 85th BirthdayDec 1, 20080 citations

Berry phase in homogeneous Kähler manifolds with linear Hamiltonians

Luis J. Boya, Askold M. Perelomov, Mariano Santander

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.

arXiv:math-ph/0111022Journal of Mathematical PhysicsNov 1, 200117 citations

THE GEOMETRICAL PROPERTIES OF BERRY’S PHASE

S. Giler, Piotr Kosiński, L. Szymanowski

Using the spectral decomposition of the Hamiltonian, we clarify the relationship of Berry’s effect with the parallel transport in the parameter space of this Hamiltonian. We derive the necessary and sufficient conditions under which this parallel transport is described by a natural, Riemannian connection. The considerations are illustrated by several examples.

International Journal of Modern Physics AApr 10, 198916 citations

A factorization property of the Berry phase

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