Chern-Simons Terms andnField in Haldane's Model for the Quantum Hall Effect without Landau Levels

It is shown that the nonzero quantized value of the Hall conductivity in Haldane's model Phys. Rev. Lett. 61 2015 1988 appears because the electron Green's function is a topologically nontrivial function of momentum and frequency. If the $\mathbf{n}$ field is introduced in this model then the same topological invariant, defined in the momentum-frequency space, gives a nonzero coefficient for the Hopf-Chern-Simons term of the $\mathbf{n}$ field. A Chern-Simons-like term, which mixes the electromagnetic and $\mathbf{n}$ fields and determines the electric charge of the Skyrmion, can also have a nonzero coefficient in the model.