Controlled expansion for transport in a class of non-Fermi liquids

Non-Fermi liquids arise when strong interactions destroy stable fermionic quasiparticles. The simplest models featuring this phenomenon involve a Fermi surface coupled to fluctuating gapless bosonic order parameter fields, broadly referred to as ``Hertz-Millis'' models. We revisit a controlled approach to Hertz-Millis models that combines an expansion in the inverse number $(N)$ of fermion species with an expansion in the deviation of the boson dynamical critical exponent $z$ from 2. The structure of the expansion is found to be qualitatively different in the quantum critical regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\ll}q$ and in the transport regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\gg}q$. In particular, correlation functions in the transport regime involve infinitely many diagrams at each order in perturbation theory. We provide an explicit and tractable recipe to classify and resum these diagrams. For the simplest Hertz-Millis models, we show that this recipe is consistent with nonperturbative anomaly arguments and correctly captures the fixed point optical conductivity as well as leading corrections from irrelevant operators. We comment on potential applications of this expansion to transport in more complicated Hertz-Millis models as well as certain beyond-Landau metallic quantum critical points.