Conformal field theories at non-zero temperature: operator product expansions, Monte Carlo, and holography

TL;DR

This paper computes the finite-temperature conductivity in 2+1D conformal field theories using operator product expansions, Monte Carlo simulations, and holography, revealing new insights into the behavior of these theories at different frequency regimes.

Contribution

It introduces a comprehensive approach combining OPE, Monte Carlo, and holography to analyze finite-temperature conductivities in 2+1D CFTs, including the role of thermal operators.

Findings

Excellent agreement between OPE predictions and Monte Carlo results for the Wilson-Fisher CFT.

Holographic methods successfully extrapolate high-frequency results to low frequencies.

Identification of the importance of thermal operators in holographic models for Wilson-Fisher CFT.

Abstract

We compute the non-zero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies much greater than the temperature, $\hbar\omega>> k_B T$, the $\omega$ dependence can be computed from the operator product expansion (OPE) between the currents and operators which acquire a non-zero 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 $\hbar \omega/(k_B T)$. 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.