The Chern states on the honeycomb and Lieb lattices

The Haldane model of the Chern insulator is considered on the Lieb and honeycomb lattices. We provide a detailed analysis of the model's ground-state phase diagram and demonstrate a scenario of the topological phase transitions in the system with a single-particle spectrum that includes flat and dispersion bands, that is realized on the Lieb lattice. We find that the Chern number of the flat band is non-zero, depending on the parameters of the model. We define the topological metal state as an intermediate state between topological insulator and trivial metal. The phase transition between topological insulator and topological metal states is accompanied by continuous changing of the Chern number, a jump of the surface charge or spin current defines the point of the topological metal-trivial metal phase transition. The results have been illustrated with numerical calculations of the model.