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## 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(), focusing specifically on scattering off composite bosons made of

two critical magnetic fluctuations. Recent study [S. A. Hartnoll et al., Phys. Rev. B 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 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(Omega) ∝ ln^3(Omega) /(Omega)^(1/3) 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/(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, σ(Omega)

scales as 1/(Omega), which is reminiscent of the behavior observed in the superconducting cuprates.