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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.
The phenomenon was independently discovered by S. Pancharatnam (1956), [2] in classical optics and by H. C. Longuet-Higgins (1958) [3] in molecular physics; it was generalized by Michael Berry in (1984). [4] It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase.
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =.The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a ...
In physics, topological order [1] is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy [2] and quantized non-abelian geometric phases of degenerate ground states. [1]
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. [1] [2] This class of groups contrasts with the abelian groups, where all pairs of group elements commute.
Non-abelian or nonabelian may refer to: Non-abelian group, in mathematics, a group that is not abelian (commutative) Non-abelian gauge theory, in physics, a gauge ...
Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model. [2] [3] The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory. [1] [4]