Optical conductivity of an interacting Weyl liquid in the collisionless regime

Optical conductivity (OC) can serve as a measure of correlation effects in a wide range of condensed-matter systems. We show that the long-range tail of the Coulomb interaction yields a universal correction to the OC in a three-dimensional Weyl semimetal $\sigma \left(\mathrm{\Omega }\right)={\sigma }_0\left(\mathrm{\Omega }\right)\left[1+\frac{1}{N+1}\right]$, where ${\sigma }_0\left(\mathrm{\Omega }\right)=N{e}_0^2\mathrm{\Omega }/\left(12hv\right)$ is the OC in the noninteracting system, with $v$ as the actual (renormalized) Fermi velocity of Weyl quasiparticles at frequency $\mathrm{\Omega }$, and $e_0$ is the electron charge in vacuum. Such universal enhancement of OC, which depends only on the number of Weyl nodes near the Fermi level ($N$), is a remarkable consequence of an intriguing conspiracy among the quantum-critical nature of an interacting Weyl liquid, marginal irrelevance of the long-range Coulomb interaction, and violation of hyperscaling in three dimensions, and can directly be measured in recently discovered Weyl as well as Dirac materials. By contrast, a local density-density interaction produces a nonuniversal correction to the OC, stemming from the nonrenormalizable nature of the corresponding interacting field theory.

Figure 1

Quantum-critical description of (a) Weyl and (b) Dirac liquid. Weyl (Dirac) semimetal can be represented as a quantum-critical point (red dots) separating electron- and hole-doped chiral Fermi liquids (FLs) (topological and normal insulators). Signatures of Weyl or Dirac quasiparticles are present within the quantum critical fan (shaded regions), where the proposed universal scaling of optical conductivity with frequency [see Eq. (1)] is operative. The crossover boundaries in (a) and (b) are respectively defined as ${\mathrm{\Omega }}^{*}\sim v{\left|n\right|}^{1/3}$ and $\left|\mathrm{\Delta }\right|$, up to interaction-driven corrections. At high frequencies ($\mathrm{\Omega }\sim E_{\mathrm{\Lambda }}$), imprints of Weyl or Dirac fermions gradually disappear and nonuniversal lattice details become important.