Transport and optics at the node in a nodal loop semimetal

We use a Kubo formalism to calculate both AC conductivity and DC transport properties of a dirty nodal loop semimetal. The optical conductivity as a function of photon energy $\mathrm{\ensuremath{\Omega}}$ exhibits an extended flat background ${\ensuremath{\sigma}}^{\mathrm{BG}}$ as in graphene provided the scattering rate $\mathrm{\ensuremath{\Gamma}}$ is small as compared to the radius of the nodal ring $b$ (in energy units). Modifications to the constant background arise for $\mathrm{\ensuremath{\Omega}}\ensuremath{\le}\mathrm{\ensuremath{\Gamma}}$ and the minimum DC conductivity ${\ensuremath{\sigma}}^{\mathrm{DC}}$, which is approached as ${\mathrm{\ensuremath{\Omega}}}^{2}/{\mathrm{\ensuremath{\Gamma}}}^{2}$ as $\mathrm{\ensuremath{\Omega}}\ensuremath{\rightarrow}0$, is found to be proportional to $\frac{\sqrt{{\mathrm{\ensuremath{\Gamma}}}^{2}+{b}^{2}}}{{v}_{F}}$ with ${v}_{F}$ the Fermi velocity. For $b=0$ we recover the known three-dimensional point node Dirac result ${\ensuremath{\sigma}}^{\mathrm{DC}}\ensuremath{\sim}\frac{\mathrm{\ensuremath{\Gamma}}}{{v}_{F}}$ while for $b>\mathrm{\ensuremath{\Gamma}}, {\ensuremath{\sigma}}^{\mathrm{DC}}$ becomes independent of $\mathrm{\ensuremath{\Gamma}}$ (universal) and the ratio $\frac{{\ensuremath{\sigma}}^{\mathrm{DC}}}{{\ensuremath{\sigma}}^{\mathrm{BG}}}=\frac{8}{{\ensuremath{\pi}}^{2}}$ where all reference to material parameters has dropped out. As $b$ is reduced and becomes of the order $\mathrm{\ensuremath{\Gamma}}$, the flat background is lost as the optical response evolves towards that of a three-dimensional point node Dirac semimetal which is linear in $\mathrm{\ensuremath{\Omega}}$ for the clean limit. For finite $\mathrm{\ensuremath{\Gamma}}$ there are modifications from linearity in the photon region $\mathrm{\ensuremath{\Omega}}\ensuremath{\le}\mathrm{\ensuremath{\Gamma}}$. When the chemical potential $\ensuremath{\mu}$ (temperature $T$) is nonzero the DC conductivity increases as ${\ensuremath{\mu}}^{2}/{\mathrm{\ensuremath{\Gamma}}}^{2}$(${T}^{2}/{\mathrm{\ensuremath{\Gamma}}}^{2}$) for $\frac{\ensuremath{\mu}}{\mathrm{\ensuremath{\Gamma}}} \left(\frac{T}{\mathrm{\ensuremath{\Gamma}}}\right)\ensuremath{\le}1$. Such laws apply as well for thermal conductivity and thermopower with coefficients of the quadratic law only slightly modified from their value in the three-dimensional point node Dirac case. However in the $\ensuremath{\mu}=T=0$ limit both have the same proportionality factor of $\sqrt{{\mathrm{\ensuremath{\Gamma}}}^{2}+{b}^{2}}$ as does ${\ensuremath{\sigma}}^{\mathrm{DC}}$. Consequently the Lorentz number is largely unmodified. For larger values of $\ensuremath{\mu}>\mathrm{\ensuremath{\Gamma}}$ away from the nodal region the conductivity shows a Drude-like contribution about $\mathrm{\ensuremath{\Omega}}\ensuremath{\approxeq}0$ which is followed by a dip in the Pauli blocked region $\mathrm{\ensuremath{\Omega}}\ensuremath{\le}2\ensuremath{\mu}$ after which it increases to merge with the flat background (two-dimensional graphene like) for $\ensuremath{\mu}b$.

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