Optical conductivity of a two-dimensional metal near a quantum-critical point: the status of the "extended Drude formula"

Abstract

The optical conductivity of a metal near a quantum critical point (QCP) is expected to depend on frequency not only via the scattering time but also via the effective mass, which acquires a singular frequency dependence near a QCP. We check this assertion by computing diagrammatically the optical conductivity, $σ' (Ω)$, near both nematic and spin-density wave (SDW) quantum critical points (QCPs) in 2D. If renormalization of current vertices is not taken into account, $σ' (Ω)$ is expressed via the quasiparticle residue $Z$ (equal to the ratio of bare and renormalized masses in our approximation) and transport scattering rate $γ_{\text{tr}}$ as $σ' (Ω)\propto Z^2 γ_{\text{tr}}/Ω^2$. For a nematic QCP ($γ_{\text{tr}}\proptoΩ^{4/3}$ and $Z\proptoΩ^{1/3}$), this formula suggests that $σ'(Ω)$ would tend to a constant at $Ω\to 0$. We explicitly demonstrate that the actual behavior of $σ' (Ω)$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, $σ' (Ω)$ diverges as $1/Ω^{2/3}$, as earlier works conjectured. In the SDW case, we consider two contributions to the conductivity: from hot spots and from"lukewarm" regions of the Fermi surface. The hot-spot contribution is not affected by vertex renormalization, but it is subleading to the lukewarm one. For the latter, we argue that a factor of $Z^2$ is again cancelled by vertex corrections. As a result, $σ' (Ω)$ at a SDW QCP scales as $1/Ω$ down to the lowest frequencies.