Transport and optics at the node in a nodal loop semimetal

We use a Kubo formalism to calculate both A.C. conductivity and D.C. transport properties of a dirty nodal loop semimetal. The optical conductivity as a function of photon energy $\Omega$, exhibits an extended flat background $\sigma^{BG}$ as in graphene provided the scattering rate $\Gamma$ is small as compared to the radius of the nodal ring $b$ (in energy units). Modifications to the constant background arise for $\Omega\le \Gamma$ and the minimum D.C. conductivity $\sigma^{DC}$ which is approached as $\Omega^2/\Gamma^2$ as $\Omega\rightarrow0$, is found to be proportional to $\frac{\sqrt{\Gamma^2+b^2}}{v_{F}}$ with $v_{F}$ the Fermi velocity. For $b=0$ we recover the known three-dimensional point node Dirac result $\sigma^{DC}\sim \frac{\Gamma}{v_{F}}$ while for $b>\Gamma$, $\sigma^{DC}$ becomes independent of $\Gamma$ (universal) and the ratio $\frac{\sigma^{DC}}{\sigma^{BG}}=\frac{8}{\pi^2}$ where all reference to material parameters has dropped out. As $b$ is reduced and becomes of the order $\Gamma$, the flat background is lost as the optical response evolves towards that of a three-dimensional point node Dirac semimetal which is linear in $\Omega$ for the clean limit. For finite $\Gamma$ there are modifications from linearity in the photon region $\Omega\le \Gamma$. When the chemical potential $\mu$ (temperature $T$) is nonzero the D.C. conductivity increases as $\mu^2/\Gamma^2$($T^2/\Gamma^2$) for $\mu/\Gamma$ $(T/\Gamma)\le 1$. For larger values of $\mu>\Gamma$ away from the nodal region the conductivity shows a Drude like contribution about $\Omega\approxeq 0$ which is followed by a dip in the Pauli blocked region $\Omega \le 2\mu$ after which it increases to merge with the flat background (two-dimensional graphene like) for $\mu< b$ and to the quasilinear (three-dimensional point node Dirac) law for $\mu> b$.