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

TL;DR

This paper investigates the optical conductivity near quantum critical points in 2D metals, revealing that vertex corrections significantly alter the expected frequency dependence and lead to divergent behavior.

Contribution

It provides a detailed diagrammatic analysis showing how vertex renormalizations modify the optical conductivity near nematic and SDW quantum critical points, correcting previous assumptions.

Findings

Optical conductivity diverges as 1/Ω^{2/3} near nematic QCP due to vertex corrections.

In SDW QCP, conductivity scales as 1/Ω at low frequencies, with vertex effects canceling Z^2.

Vertex renormalization crucially affects the frequency dependence of optical responses.

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, $\sigma' (\Omega)$, near both nematic and spin-density wave (SDW) quantum critical points (QCPs) in 2D. If renormalization of current vertices is not taken into account, $\sigma' (\Omega)$ is expressed via the quasiparticle residue $Z$ (equal to the ratio of bare and renormalized masses in our approximation) and transport scattering rate $\gamma_{\text{tr}}$ as $\sigma' (\Omega)\propto Z^2 \gamma_{\text{tr}}/\Omega^2$. For a nematic QCP ($\gamma_{\text{tr}}\propto\Omega^{4/3}$ and $Z\propto\Omega^{1/3}$), this formula suggests that $\sigma'(\Omega)$ would tend to a constant at $\Omega \to 0$. We explicitly demonstrate that the actual behavior of $\sigma' (\Omega)$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, $\sigma' (\Omega)$ diverges as $1/\Omega^{2/3}$, 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, $\sigma' (\Omega)$ at a SDW QCP scales as $1/\Omega$ down to the lowest frequencies.