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In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.
Behavior of the curvature function (Berry connection) for the Kitaev model (top panel) as a function of a parameter change in parameter space (bottom panel). The vertical red lines depict topological phase boundaries. Across the topological phase transitions, the Berry connection diverges and flips sign around the high-symmetry points k=0.
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.
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.
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 ...
The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium. A convenient way to understand the curvature comes from an ordinary differential equation, first ...
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus , Newton's method (also called Newton–Raphson ) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} .
The complex beam parameter is usually used in ray transfer matrix analysis, which allows the calculation of the beam properties at any given point as it propagates through an optical system, if the ray matrix and the initial complex beam parameter is known.