Quantum critical charge response from higher derivatives in holography

We extend the range of possibilities for the charge response in the quantum critical regime in 2 + 1D using holography, and compare them with field theory and recent quantum Monte Carlo results. We show that a family of (infinitely many) higher derivative terms in the gravitational bulk leads to behavior far richer than what was previously obtained. For example, we prove that the conductivity becomes unbounded, undermining previously obtained constraints. We further find a nontrivial and infinite set of theories that have a self-dual conductivity. Particle-vortex or S duality plays a key role; notably, it maps theories with a finite number of bulk terms to ones with an infinite number. Many properties, such as sum rules and stability conditions, are proven.

Figure 1

(a) Sketch of the subspace of holographic theories describing the charge response of CFTs in 2 + 1D. It encompasses any action with $n_{\mathrm{term}}$ terms containing two field strengths coupled to curvature tensors. The arrows indicate S duality. The central regions (red) have a self-dual, $\omega$-independent conductivity $\sigma$. The star corresponds to the R-charge response of superconformal Yang-Mills for $N=\lambda =\infty$. (b) Allowed asymptotics for the dc conductivity, diffusion constant $D$ and susceptibility $\chi$; ${\gamma }_1$ parameterizes a six-derivative term.

Figure 2

Real part of the conductivity for $\gamma =0$, with the corresponding $({\gamma }_1,{\gamma }_2)$ indicated in the inset. Inset: Portion of the allowed parameter space for $\gamma =0$, and contour plot of $\sigma \left(0\right)$, where blue (red) indicates small (large) values. The green dashed lines in the main figure and inset are associated with the self-dual conductivity.

Figure 3

Comparing the conductivity of (a) the quantum critical O($N\to \infty$) model (with small broadening; see text) to that of (b) the six-derivative holographic action with $(\gamma ,{\gamma }_1,{\gamma }_2)=(0,-1,0)$. The inverse conductivities (dashed line) also show similarities. Insets: Analytic structure of $\sigma \left(w\right)$ in complex $w$ plane: crosses (circles) represent poles (zeros) [the line in (a) is a branch cut].

Figure 4

(a) The markers correspond to the imaginary-frequency conductivity obtained from large-scale quantum Monte Carlo simulations on the O(2) QCP [13]. The blue (dot-dashed) and red (solid) curves are the fits to the data using the Weyl action [13] or the one including six-derivative terms in addition, respectively. The values of the fitting parameters are shown. (b) Corresponding real-frequency behavior.