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

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. On the other hand, the quasiparticle residue $Z$, no matter how singular, does not appear in the conductivity as the latter probes quasiparticles rather than bare electrons. In local theories of QCPs, however, the ratio of band and renormalized masses, $m^*/m_b$, coincides with $1/Z$, and it is not straightforward to separate the two quantities.In this work, we use a direct diagrammatic approach and compute the optical conductivity, $\sigma' (\Omega)$,
near a 2D nematic QCP, using the local approximation in which $Z = m_b/m^*$. If renormalization of current vertices is not taken into account, $\sigma' (\Omega)$ is expressed via $Z = m_b/m^*$ and the transport scattering rate $\gamma_{\text{tr}}$ as $\sigma' (\Omega) \propto Z^2 \gamma_{\text{tr}}/\Omega^2$. We explicitly demonstrate that the renormalization of the current vertices cancels out a factor of $Z^2$. As a result, $\sigma' (\Omega)$ diverges as $1/\Omega^{2/3}$, as earlier works conjectured. We also consider conductivity near an antiferromagnetic QCP in a metal.