We compute the nonzero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 space-time dimensions. At frequencies much greater than the temperature, $\hslash \omega \gg k_{B}T$, the $\omega$ dependence can be computed from the operator product expansion (OPE) between the currents and operators, which acquire a nonzero expectation value at $T>0$. Such results are found to be in excellent agreement with quantum Monte Carlo studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are also obtained in vector $1/N$ expansions. We match these large $\omega$ results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small $\hslash \omega /\left(k_{B}T\right)$. Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large $\omega$ behavior for a large class of CFTs. However, for the Wilson-Fisher CFT, a relevant “thermal” operator must also be considered, and then consistency with the Monte Carlo results is obtained without a previously needed ad hoc rescaling of the $T$ value. We also establish sum rules obeyed by the conductivity of a wide class of CFTs.

• Received 23 September 2014
• Revised 17 November 2014

DOI:https://doi.org/10.1103/PhysRevB.90.245109