Quantum critical response: from conformal perturbation theory to holography

TL;DR

<p>We study dynamical response functions near quantum critical points described by a conformal field theory (CFT), allowing for finite temperature and detuning by a relevant operator. Using conformal perturbation theory and holography, we derive a universal high-frequency expansion for two-point functions, governed by UV CFT data, and establish nonperturbative sum rules that constrain low-frequency behavior. The two approaches are shown to be in exact agreement, and we extend the analysis to holographic Lifshitz theories, where high-frequency behavior remains similar despite weaker symmetry constraints. We also present full-frequency holographic results, including minimal models and Reissner–Nordström backgrounds, and demonstrate how the high-frequency data determine all leading coefficients in the UV, with explicit expressions for scalar, current, and stress-tensor probes. These results provide robust, testable constraints for quantum critical dynamics that can be probed in quantum Monte Carlo simulations and cold-atom experiments.</p>

Abstract

We discuss dynamical response functions near quantum critical points, allowing for both a finite temperature and detuning by a relevant operator. When the quantum critical point is described by a conformal field theory (CFT), conformal perturbation theory and the operator product expansion can be used to fix the first few leading terms at high frequencies. Knowledge of the high frequency response allows us then to derive non-perturbative sum rules. We show, via explicit computations, how holography recovers the general results of CFT, and the associated sum rules, for any holographic field theory with a conformal UV completion -- regardless of any possible new ordering and/or scaling physics in the IR. We numerically obtain holographic response functions at all frequencies, allowing us to probe the breakdown of the asymptotic high-frequency regime. Finally, we show that high frequency response functions in holographic Lifshitz theories are quite similar to their conformal counterparts, even though they are not strongly constrained by symmetry.

Figures (9)

• Figure 1: Schematic phase diagram of a theory with a (conformal) quantum critical point separating two phases at zero temperature. The non-thermal detuning parameter $h$ must be set to zero at the critical point. At finite temperature $T$, when $T\gtrsim |h|^{\frac{1}{D-\Delta}}=|h|^\nu$, there is a window in between the two phases where the system behaves qualitatively like the quantum critical point $h=0$, detuned only by temperature. In this paper, we demonstrate that certain fingerprints in the dynamical response functions of the quantum critical point can extend throughout this entire phase diagram.
• Figure 2: The scalar field with $D=4$ as a function of $u$ for various scaling dimensions $\Delta$ with $\Phi_1$ fixed to 1.
• Figure 3: Scalar two point function $\langle \mathcal{O}_0(-\Omega) \mathcal{O}_0(\Omega)\rangle$ in $D=3$ at finite temperature, with scaling dimensions $\Delta_0 = 2$ and $\Delta = 2$. We have set $\alpha_Y\alpha_W =1$. Left: full frequency response for the scalar two point function. Right: comparison between the analytic prediction for the leading correction to the linear response plotted against the numerically calculated response with the leading term removed (plotted in a log-log plot).
• Figure 4: The Euclidean (left) and real (right) AC conductivity for $D=3$ and $\Delta=2$ in the critical theory ($h=0$) for various choices of interaction strength $\alpha_W\alpha_Z$.
• Figure 5: The real part of the AC conductivity for $D=4$ and $\Delta=1$ (left) and $\Delta=3$ (right), in the critical theory ($h=0$), for various choices of coupling constants $\alpha_W \alpha_Z$. At large $\omega$, $\mathrm{Re}(\sigma(\omega)) \approx \sigma_\infty \omega$, with $\sigma_\infty = \frac{\pi}{2} \tilde{\mathcal{C}}_{JJ}$ given in (\ref{['eq:2resigma']}).