Gifts from anomalies: optical conductivity of metallic quantum critical points and other exact results

The optical conductivity of strange metals appears to display a "scale-invariant'' form as a function of frequency and temperature. Although this is traditionally interpreted as a signature of quantum criticality, such an interpretation requires careful scrutiny, as it implies that the infrared renormalization group fixed-point theory that controls the quantum criticality must have a non-trivial frequency dependence of the optical conductivity. In this talk, I will present our results on optical transport in a model of a Landau ordering transition in metallic system, where strong fluctuations of a bosonic order parameter destroy the quasiparticles. We demonstrate that exact, non-perturbative results can be obtained for optical transport, and some other quantities, in such "Hertz-Millis" theories, by leveraging the large emergent symmetry and anomaly structure. In particular, we show that in the infrared limit, the boson self energy at zero wave vector, q=0, is a constant independent of frequency, and the real part of the optical conductivity, σ(ω), is purely a delta function Drude peak with no other corrections. Therefore, further frequency dependence in the boson self energy or optical conductivity can only come from irrelevant operators in a clean system. By contrast, in a modified version of the model with N flavors that has been discussed in past literature, we show that, in the large-N limit, there is a non-trivial optical conductivity in the infrared fixed point.