Tight-binding theory of spin-orbit coupling in graphynes

We investigate the effects of Rashba and intrinsic spin-orbit couplings in graphynes. First, we develop a general method to address spin-orbit couplings within the tight-binding theory. Then, we apply this method to $\alpha$, $\beta$, and $\gamma$-graphyne, and determine the SOC parameters in terms of the microscopic hopping and on-site energies. We find that for $\alpha$-graphyne, as in graphene, the intrinsic SOC opens a non-trivial gap, whereas the Rashba SOC splits each Dirac cone into four. In $\beta$ and $\gamma$ graphyne, the Rashba SOC can lead to a Lifshitz phase transition, thus transforming the zero-gap semiconductor into a gapped system or vice versa, when pairs of Dirac cones annihilate or emerge. The existence of internal (within the benzene ring) and external SOC in these compounds allow us to explore a myriad of phases not available in graphene.