Vanishing of anomalous quantum Hall and quantum spin Hall phase in band deformed insulators.

We look for distinct topological phases of a two dimensional honeycomb lattice, one in which the time reversal symmetry (TRS) is violated, and in another it is preserved. In these systems, the vanishing of the topological phases can be induced via a band deformation that may be caused by varying the nearest neighbour hopping parameters. In particular, among the neighbouring sites, one may continuously vary one of them (say, t₁), while keeping the other two hopping terms (say, t) fixed. For the TRS broken case, where there is a topological gap in the system caused by complex second neighbour hopping (Haldane model), the gap vanishes at t₁=2t, the so called semi-Dirac limit. As t₁ becomes larger than 2t, a gap reopens in the spectrum, which however can be shown to be a trivial gap and the system behaves like a band insulator. We study such a topological phase transition via the disappearance of the anomalous Hall conductivity and the Chern number. In the TRS invariant case, which can be considered as two copies of the Haldane model, one for each spin, there occurs a similar vanishing of the quantum spin Hall phase that can be captured through vanishing of the topological invariant (the Ζ₂ invariant) in the semi-Dirac limit, beyond which the system demonstrates dispersion same as that of a trivial insulator. Summarizing, we show that a band deformation in a honeycomb lattice is capable of inducing a phase transition irrespective of whether TRS is broken or preserved.