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∗/mb, 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, σ′(Ω), near two-dimensional (2D) nematic and spin-density wave (SDW) QCPs, using the local approximation in which Z=mb/m∗. If renormalization of current vertices is not taken into account, σ′(Ω) is expressed via Z=mb/m∗ and the transport scattering rate γtr as σ′(Ω)Z2γtr/Ω2. For a nematic QCP (γtrΩ4/3 and ZΩ1/3), this formula suggests that σ′(Ω) would tend to a constant at Ω→0. We explicitly demonstrate that the actual behavior of σ′(Ω) is different due to strong renormalization of the current vertices, which cancels out a factor of Z2. As a result, σ′(Ω) diverges as 1/Ω2/3, 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 Z2 is again canceled by vertex corrections. As a result, σ′(Ω) at a SDW QCP scales as 1/Ω down to the lowest frequencies, up to possible multiplicative logarithmic factors.
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We thank S. Hartnoll, Y.B. Kim, and S. Sachdev for fruitful discussions and the anonymous referee for highlighting the issue of the difference between the mass renormalization factor and quasiparticle residue and for rightly questioning our initial results for the Drude peak at a QCP. This work was supported by the NSF DMR-1523036 (A.V.C.). We acknowledge the hospitality of the Kavli Institute for Theoretical Physics (KITP) in Santa Barbara, CA. KITP is supported by the NSF via Grant No. NSF PHY11-25915.