Signatures of merging Dirac points in optics and transport

We consider the optical and transport properties in a model two-dimensional Hamiltonian which describes the merging of two Dirac points. At low energy, in the presence of an energy gap parameter $\Delta$, there are two distinct Dirac points with linear dispersion, these are connected by a saddle point at higher energy. As $\Delta$ goes to zero, the two Dirac points merge and the resulting dispersion exhibits semi-Dirac behaviour which is quadratic in the $x$-direction ("nonrelativistic") and linear the $y$-direction ("relativistic").In the clean limit for each direction ($x,y$) the contribution of the intraband and interband optical transitions are both given by universal functions of photon energy $\Omega$ and chemical potential $\mu$ normalized to the energy gap. We provide analytic formulas for both small and large $\Omega/2\Delta$ and $\mu/\Delta$ limits. These define, respectively, Dirac and semi-Dirac-like regions. For $\Omega/2\Delta$ and $\mu/\Delta$ of order one, there are deviations from these asymptotic behaviors. Considering optics and also transport, such as dc conductivity, thermal conductivity and the Lorenz number, such deviations provide signatures of the evolution from the Dirac to the semi-Dirac regime as the gap $\Delta$ is varied.