Optical conductivity of a two-dimensional metal at the onset of spin-density-wave order

We consider the optical conductivity of a clean two-dimensional metal near a quantum spin-density-wave transition. Critical magnetic fluctuations are known to destroy fermionic coherence at "hot spots" of the Fermi surface but coherent quasiparticles survive in the rest of the Fermi surface. A large part of the Fermi surface is not really "cold" but rather "lukewarm" in a sense that coherent quasiparticles in that part survive but are strongly renormalized compared to the non-interacting case. We discuss the self-energy of lukewarm fermions and their contribution to the optical conductivity, $\sigma(\Omega)$, focusing specifically on scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S.A. Hartnoll et al., Phys. Rev. B {\bf 84}, 125115 (2011)] found that composite scattering gives the strongest contribution to the self-energy of lukewarm fermions and suggested that this may give rise to non-Fermi liquid behavior of the optical conductivity at the lowest frequencies. We show that the most singular term in the conductivity coming from self-energy insertions into the conductivity bubble, $\sigma'(\Omega)\propto \ln^3\Omega/\Omega^{1/3}$, is canceled out by the vertex-correction and Aslamazov-Larkin diagrams. However, the cancelation does not hold beyond logarithmic accuracy, and the remaining conductivity still diverges as $1/\Omega^{1/3}$. We further argue that the $1/\Omega^{1/3}$ behavior holds only at asymptotically low frequencies, well inside the frequency range affected by superconductivity. At larger $\Omega$, up to frequencies above the Fermi energy, $\sigma'(\Omega)$ scales as $1/\Omega$, which is reminiscent of the behavior observed in the superconducting cuprates.