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 noninteracting case. We discuss the self-energy of lukewarm fermions and their contribution to the optical conductivity σ(Ω), focusing specifically on scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S. A. Hartnoll, Phys. Rev. B 84, 125115 (2011)PRBMDO1098-012110.1103/PhysRevB.84. 125115] found that composite scattering gives the strongest contribution to the self-energy of lukewarm fermions and suggested that this may give rise to a 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 σ′(Ω)ln3Ω/ Ω1/3 is canceled out by the vertex-correction and Aslamazov-Larkin diagrams. However, the cancellation does not hold beyond logarithmic accuracy, and the remaining conductivity still diverges as 1/Ω1/3. We further argue that the 1/Ω1/3 behavior holds only at asymptotically low frequencies, well inside the frequency range affected by superconductivity. At larger Ω, up to frequencies above the Fermi energy, σ′(Ω) scales as 1/Ω, which is reminiscent of the behavior observed in the superconducting cuprates.