Transport and optics at the node in a nodal loop semimetal

Abstract

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 $\Omega \to 0$, 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 $\mu >b$.

11 pages, 9 figures, accepted in Phys. Rev. B

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• Optical conductivity of the threefold Hopf semimetal
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