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, ${\sigma }^{\prime }(\Omega )$, near both nematic and spin-density wave (SDW) quantum critical points (QCPs) in 2D. If renormalization of current vertices is not taken into account, ${\sigma }^{\prime }(\Omega )$ 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 ${\sigma }^{\prime }(\Omega )\propto Z^2{\gamma }_{\text{tr}}/{\Omega }^2$. For a nematic QCP (${\gamma }_{\text{tr}}\propto {\Omega }^{4/3}$ and $Z\propto {\Omega }^{1/3}$), this formula suggests that ${\sigma }^{\prime }(\Omega )$ would tend to a constant at $\Omega \to 0$. We explicitly demonstrate that the actual behavior of ${\sigma }^{\prime }(\Omega )$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, ${\sigma }^{\prime }(\Omega )$ 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, ${\sigma }^{\prime }(\Omega )$ at a SDW QCP scales as $1/\Omega$ down to the lowest frequencies.

arXiv comment14 pages, 6 figures

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• Dynamical susceptibility near a long-wavelength critical point with a nonconserved order parameter
• Theory of shot noise in strange metals
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