Sobre a topologia no efeito Hall quântico: um passeio através da geometria diferencial e topologia

Abstract

If we have a system where the Hamiltonian’s self-state is non-degenerate, after a cyclic adiabatic evolution it will return to its initial state plus a phase. This is due to the dynamics of the system and also due to the path traced in R, where R is parameter space that retains memory of its path referring to the system of interest. If we consider this trajectory a closed curve C, we can say that geometric phase is the flux of a field that crosses a given surface bounded by C and Berry’s phase can be rewritten using Stokes’s theorem. Finally we have a local manifestation of invariance of the properties of wave functions in the parameter space which is called Berry curvature. In the context of semiconductors, these developments have changed our conception of the usual band structure theory, which now naturally incorporates issues of topology, dispersion relations, and their self-states. In mathematics, topology studies properties that are not affected by continuous deformations through a topological variety M, and since these properties are somewhat constant by continuous transformations we can say that these are topological invariants.In particular, Thouless, Kohmoto, Nightingale, den Nijs, they showed that in the presence of intense magnetic fields, the quantization oh the quantum Hall conductivity can be understood in terms of a topological invariant, the Chern number; which is calculated by integral of the Berry curvature along the Brillouin zone, and to calculate this integral a beautiful differential theorem of geometry is used, which is primarily due initially to Gauss and Bonnet, and later to the Chinese Chern. Thus, because this theorem connects the geometry of a surface with its topology, we can derive information from it simply by looking at its topology, from a magnitude called genus g, which is invariant by continuous transformations. The genus which is roughly the number of holes contained in a closed surface.