Vanishing of the quantum spin Hall phase in a semi-Dirac Kane Mele model

We study the vanishing of the topological properties of a quantum spin Hall insulator induced by a deformation of the band structure that interpolates between the Dirac and the semi-Dirac limits of a tight-binding model on a honeycomb lattice. The above scenario is mimicked in a simple model, where there exists a differential hopping along one of the three neighbours (say, $t_1$) compared to the other two (say, $t$). For $t_1 = t$, the properties of the quantum spin Hall phase is described by the familiar Kane Mele model, while $t<t_1<2t$ denotes a situation in which the spin resolved bands are continuously deformed. $t_1 = 2t$ represents a special case which is called as the semi-Dirac limit. Here, the spectral gaps between the conduction and the valence bands vanish. A closer inspection of the properties of such a deformed system yields insights on a topological phase transition occurring at the semi-Dirac limit, which continues to behave as a band insulator for $t_1>2t$. We demonstrate the evolution of the topological phase in presence of the Rashba and intrinsic spin-orbit couplings via computing the electronic band structure, edge modes in a nanoribbon and the $\mathbb{Z}_2$ invariant. The latter aids in arriving at the phase diagram which conclusively shows vanishing of the topological phase in the semi-Dirac limit. Further we demonstrate in gradual narrowing down of the plateau in the spin Hall conductivity, which along with a phase diagram provide robust support on the vanishing of the $\mathbb{Z}_2$ invariant and hence the quantum spin Hall phase.