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 $\mathrm{\Omega }$, exhibits an extended flat background ${\sigma }^{BG}$ as in graphene provided the scattering rate $\mathrm{\Gamma }$ is small as compared to the radius of the nodal ring $b$ (in energy units). Modifications to the constant background arise for $\mathrm{\Omega }\le \mathrm{\Gamma }$ and the minimum D.C. conductivity ${\sigma }^{DC}$ which is approached as ${\mathrm{\Omega }}^2\mathrm{/}{\mathrm{\Gamma }}^2$ as $\mathrm{\Omega }\to 0$, is found to be proportional to $\frac{\sqrt{{\mathrm{\Gamma }}^2+b^2}}{v_F}$ with $v_F$ the Fermi velocity. For $b=0$ we recover the known three-dimensional point node Dirac result $\sigma^{DC}\sim \frac{\Gamma}{v_{F}}$whilefor$b>\Gamma$,$\sigma^{DC}$ becomes independent of $\mathrm{\Gamma }$ (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 $\mathrm{\Gamma }$, the flat background is lost as the optical response evolves towards that of a three-dimensional point node Dirac semimetal which is linear in $\mathrm{\Omega }$ for the clean limit. For finite $\mathrm{\Gamma }$ there are modifications from linearity in the photon region $\mathrm{\Omega }\le \mathrm{\Gamma }$. When the chemical potential $\mu$ (temperature $T$) is nonzero the D.C. conductivity increases as ${\mu }^2\mathrm{/}{\mathrm{\Gamma }}^2$($T^2\mathrm{/}{\mathrm{\Gamma }}^2$) for $\mu \mathrm{/}\mathrm{\Gamma }$ $(T\mathrm{/}\mathrm{\Gamma })\le 1$. For larger values of $\mu >\mathrm{\Gamma }$ away from the nodal region the conductivity shows a Drude like contribution about $\mathrm{\Omega }\approxeq 0$ which is followed by a dip in the Pauli blocked region $\mathrm{\Omega }\le 2\mu$ 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 $\mu >b$.