Chern insulator with large Chern numbers. Chiral Majorana fermion liquid

In the framework of Hofstadter`s approach we provide a detailed analysis of a realization of exotic topological states such as the Chern insulator with large Chern numbers. In a transverse homogeneous magnetic field a one-particle spectrum of fermions transforms to an intricate spectrum with a fine topological structure of the subbands. In a weak magnetic field $H$ for a rational magnetic flux, a topological phase with a large Chern number is realized near the half filling. There is an abnormal behavior of the Hall conductance ${\sigma_{xy}\simeq \frac{e}{2 H}}$. At half-filling, the number of chiral Majorana fermion edge states increases sharply forming a new state, called the chiral Majorana fermion liquid.