Adiabatic Process and Chern Numbers
نویسندگان
چکیده
Recently applications of the Chern number have attracted a lot of interest. The notable example is the quantum Hall effect. It can be argued that the precise quantization of Hall conductance should be related to topological nature of the systems. In fact the quantization of the Hall conductance in two-dimensional periodic potentials was proved by using the Kubo formula by Thouless, Kohmoto, Nightingale, and den Nijs1. It was shown later by one of us2 that the quantization is due to the topological nature of the problem and Hall conductance is given by Chern numbers in the theory of fiber bundle of differential geometry. The base manifold is T, which is a magnetic Brillouin zone, and the fiber is wavefunctions. Niu, Thouless, and Wu proposed an interesting idea of twisted boundary conditions which yield Chern numbers even in systems without translation symmetry3. Other examples include quantum Hall effect in three dimensions4, anomalous quantum Hall effect in ferromagnets5, spin Hall effect in vortex states 6, and Thouless pumping 7. In this letter we propose a new method involving Chern numbers to have a unified point of view in adiabatic quantum process. We shall apply this theory to the examples above. In addition we shall discuss AC Josephson effect and spin Hall effect in semiconductors10 and obtain a new perspective to these problems.
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