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In this case, the Berry phase corresponding to any given path on the unit sphere in magnetic-field space is just half the solid angle subtended by the path. The integral of the Berry curvature over the whole sphere is therefore exactly 2 π {\displaystyle 2\pi } , so that the Chern number is unity, consistent with the Chern theorem.
There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic.
The Hannay angle is defined in the context of action-angle coordinates.In an initially time-invariant system, an action variable is a constant. After introducing a periodic perturbation (), the action variable becomes an adiabatic invariant, and the Hannay angle for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the ...
This is termed the integral quantum Hall effect. These oscillations exhibit a phase shift of π, known as Berry's phase, [10] [3] which is due to the zero effective mass of carriers near the Dirac points. [48] Despite this zero effective mass, the temperature dependence of the oscillations indicates a non-zero cyclotron mass for the carriers. [10]
Trigonal bipyramidal molecular shape ax = axial ligands (on unique axis) eq = equatorial ligand (in plane perpendicular to unique axis). The Berry mechanism, or Berry pseudorotation mechanism, is a type of vibration causing molecules of certain geometries to isomerize by exchanging the two axial ligands (see the figure) for two of the equatorial ones.
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
Wigner quasiprobability distribution of squeezed states, for varying amount of phase shift and displacement. When the phase shift is zero (leftmost column), the peak-to-peak amplitude of oscillation is less uncertain, but the phase (the time at which crosses the midpoint) is more uncertain. The opposite is true when the phase shift is half a ...
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