Optical response in Weyl semimetal in model with gapped Dirac phase

We study the optical properties of Weyl semimetal (WSM) in a model which features, in addition to the usual term describing isolated Dirac cones proportional to the Fermi velocity $v_{F}$, a gap term $m$ and a Zeeman spin-splitting term $b$ with broken time reversal symmetry. Transport is treated within Kubo formalism and particular attention is payed to the modifications that result from a finite $m$ and $b$. We consider how these modifications change when a finite residual scattering rate $\Gamma$ is included. For $\Gamma<m$ the A.C. conductivity as a function of photon energy $\Omega$ continues to display the two quasilinear energy regions of the clean limit for $\Omega$ below the onset of the second electronic band which is gapped at ($ m+b $). For $\Gamma$ of the order $m$ little trace of two distinct linear energy scales remain and the optical response has evolved towards that for $m=b=0$. Although some quantitative differences remain there are no qualitative differences. The magnitude of the D.C. conductivity $\sigma^{DC}(T=0)$ at zero temperature ($T=0$) and chemical potential ($\mu=0$) is altered. While it remains proportional to $\Gamma$ it becomes inversely dependent on an effective Fermi velocity out of the Weyl nodes equal to $v_{F}^\ast=v_{F}\sqrt{b^2-m^2}/b$ which decreases strongly as the phase boundary between Weyl semimetal and gapped Dirac phase (GDSM) is approached at $b=m$. The leading term in the approach to $\sigma^{DC}(T=0)$ for finite $T/\Gamma$, $\mu/\Gamma$ and $\Omega/\Gamma$ is found to be quadratic. The coefficient of these corrections tracks closely the $b/m$ dependence of the $\mu=T=\Omega=0$ limit with differences largest near to the WSM-GDSM boundary.