Using the energy Green's function formulation proposed by Niu 1 for particle densities, we construct and clarify the nature of the topological invariant assigned to the Hall conductance in the Hall system of 2-dimensional noninteracting electron gas; we identify this topological quantum number explicitly as the first Chern number of a complex vector bundle over a 2-torus parametrized by the magnetic potential (a1, a2); the fibres are finite dimensional spaces spanned by eigenfunctions of the system with energy eigenvalues below the Fermi energy. Other cases can be treated by a similar procedure, namely, by recognizing that some physical quantities are integrals of curvatures defined on a nontrivial finite dimensional complex bundle. Therefore, in suitable units, they take integer values. We treat, as an example, the electron density response to a dilation of a periodic potential. The integer in this case is the number of Bloch bands. The quantization of the Hall conductance and density response is also shown in the presence of disorder.
Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves
