geometric phase

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Geometric phase

A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space. Under various aspects, this concept occurs in geometry, astronomy, classical mechanics, and quantum theory. In geometry it is known as holonomy. In quantum theory it is known as Berry's phase, after M. Berry, who isolated the concept (which was already known in special cases) and explained its wide-ranging signi-ficance.

A system is envisioned whose possible states can be visualized as points in a suitable abstract space. At the same time, the system has some position or orientation in another space. A history of internal states can be represented by a curve in the first space; and the effect of this history on the disposition of the system, by a curve in the second space. The mapping between these two curves is described by the geometric phase. Especially interesting is the case when a closed curve (cycle) in the first space maps onto an open curve in the second, for then there is no net change in internal state, yet the disposition of the system with respect to the outside world is altered.

The power of the geometric phase ideas is that they make it possible, in complex dynamical problems, to find some simple universal regularities without having to solve the complete equations. Significant uses of these ideas include demonstrations of the fractional electric charge and quantum statistics of the quasiparticles in the quantum Hall effect, and of the occurrence of anomalies in quantum field theory. See Anyons, Hall effect, Quantum field theory

geometric phase

[‚jē·ə‚me·trik ′fāz]
A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space.
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In second section using model Hamiltonian together with Berry phase approach we obtained explicit expression of Berry curvature, spin and valley split form of Hall conductivity and total valley and spin Hall conductivity.
The nonlinear susceptibility control is based on a novel mechanism that involves Berry phase in the nonlinear optical regime, which arises from the coupling between the spin of the photons and the orientation of the individual optical antennas.
Berry phase physics and spin-scattering in time-dependent magnetic fields has been studied by Sarah Maria Schroeter [6].
Baer, a leading authority on molecular scattering theory and electronic non-diabetic processes, addresses the well-know deficiency in conical intersections, covering the mathematics, diabetization on the topological matrix, model studies, studies of molecular systems, degeneracy points and coupling terms as poles, the molecular field, open phase and berry phase for molecular systems, and extended Born-Oppenheimer approximations.
The latter is termed Berry phase or more general geometric phase in contrast to the former dynamical phase [[phi].sub.d].
It covers topics ranging from band theory and semiconductor physics to quantum dots, Berry phases, and more, and offers insights into both theory and experiment.
He also points out that "the entire discipline of condensed matter is roughly ten percent older than when the first edition was written, so adding some new topics seemed appropriate." These new topics--chosen because of increasing recognition of their importance--include graphene and nanotubes, Berry phases, Luttinger liquids, diffusion, dynamic light scattering, and spin torques.