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

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, $\hbarω>> k_B T$, the $ω$ 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 $ω$ results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small $\hbar ω/(k_B T)$. Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large $ω$ 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.

Figures (7)

• Figure 1: QMC results (open circles) at $K_c=0.3330671$ with $\mu=0$ for the frequency dependent conductivity $\sigma(i\omega_n)$. All results have first been extrapolated to $L\to\infty$ and subsequently to $T\to 0$ ($L_\tau\to\infty$). The solid blue line shows a fit to the QMC data for $n=1,\ldots,7$ of the form $2\pi\sigma/\sigma_Q=0.3605+0.054/n^{1.533}-0.01/n^3$ with $n=\omega_n/(2\pi T)$ the Matsubara index. The dashed blue line is the continuation of the fitted form to $n>7$.
• Figure 2: QMC results (open circles) at $K_c=0.3330671$ for $\langle \mathcal{O}_g \rangle$ in the limit $L\to\infty$ as a function of $L_\tau$. The solid red line indicates a fit to the QMC data of the indicated form.
• Figure 3: QMC results (open circles) at $K_c=0.3330671$ for $\langle \mathcal{O}_g(0) \mathcal{O}_g(\tau) \rangle_{L_\tau}-\langle \mathcal{O}_g\rangle^2_{L_\tau}$ in the limit $L\to\infty$ as a function of the imaginary time, $\tau$. Results are shown for different values of $L_\tau$. The dashed red line indicates the $L_\tau\to\infty$ limit of $\langle \mathcal{O}_g(0) \mathcal{O}_g(\tau) \rangle_{L_\tau}-\langle \mathcal{O}_g\rangle^2_{L_\tau} \ \to\ 0.0122 \tau^{-(6-2/\nu)}$ with $\nu=0.6714$.
• Figure 4: a) Holographic fit (line) to Quantum Monte Carlo data for the conductivity of a model in its O$(2)$ quantum critical regime (dots). The holographic parameters are: $\Delta = 3/2, a\alpha = 0.6$. b) The corresponding conductivity on the real (Minkowski) frequency axis (solid line). The dashed line corresponds to the holographic fit obtained in Ref.krempa_nature, where an ad hoc rescaling of temperature was needed.
• Figure 5: The location of the small-frequency poles (crosses) and zeros (circles) of the holographic conductivity $\sigma(\omega)$ in the complex frequency plane. The parameters used are the same as those use to fit the O(2) QCP, see Fig. \ref{['fig:qmc_sig']}. The dominant, purely damped pole is denoted by D-QNM, where QNM stands for quasinormal mode.