Emergent scale invariance of disordered horizons

TL;DR

The paper demonstrates that marginally relevant quenched disorder in a holographic CFT drives the finite-temperature horizon to an IR fixed point with emergent scale invariance, evidenced by entropy scaling $s \sim T^{(d-1)/z}$ and a numerically computed dynamical exponent $z = 1 + \frac{1}{2} \pi^{\frac{d}{2}-1} \Gamma\left(\frac{d}{2}\right) \bar{V}^2 + O(\bar{V}^4)$. The authors construct both perturbative and fully backreacted black hole solutions in AdS$_{d+1}$ with Gaussian boundary disorder, showing that the averaged IR geometry captures the correct scaling despite inhomogeneity. They demonstrate that the entropy-temperature relation derived from the averaged metric agrees with that of full disorder realizations, supporting the use of averaged geometries to characterize disordered fixed points. Numerically, the DeTurck-based solutions in $d=2$ and $d=3$ corroborate the analytic prediction and reveal that disorder leaves a persistent imprint on the horizon structure and transport properties, signaling a novel IR quantum critical regime with broken momentum conservation. Overall, the work provides concrete holographic evidence for emergent scale invariance in disordered horizons and clarifies how averaged bulk observables reflect IR critical behavior.

Abstract

We construct planar black hole solutions in AdS_3 and AdS_4 in which the boundary CFT is perturbed by marginally relevant quenched disorder. We show that the entropy density of the horizon has the scaling temperature dependence s \sim T^{(d-1)/z} (with d=2,3). The dynamical critical exponent z is computed numerically and, at weak disorder, analytically. These results lend support to the claim that the perturbed CFT flows to a disordered quantum critical theory in the IR.

Figures (4)

• Figure 1: Disordered sources: Plot (a) shows a scalar source $\Phi_1$ as a function of $x_1\,k_0$ at the boundary. Plot (b) is a density plot of $\Phi_1$, now in $d=3$, as a function of boundary directions $x_1\,k_0$ and $x_2\,k_0$. In both cases we have chosen $\bar{V}=0.1$. The characteristic width of the peaks in these plots is determined by the short distance cutoff, $\Delta x |_\text{peak} \sim \pi/k_0$.
• Figure 2: Emergence of an IR dynamical scaling exponent. Plot (a) shows the logarithmic derivative of the entropy for several values of $\bar{V}$ in $d=2$. These plots have $N=50$. From top to bottom, we have $\bar{V}=0.1,0.2,\ldots,1.0$. Plot (b) shows the logarithmic derivative of the entropy for $\bar{V}=0.1$ in $d=3$. This plot has $N=5$.
• Figure 3: Disordered horizons. Plot (a) shows the scalar field $\Phi_\mathcal{H}$ evaluated at the horizon as a function of $x_1\,k_0$. Plot (b) is a density plot of $\Phi_\mathcal{H}$, now in $d=3$, as a function of boundary directions $x_1\,k_0$ and $x_2\,k_0$. The sources for these solutions are those shown in Fig. \ref{['figs:0']}. Plot (a) is at temperature $T/k_0 = 0.0478$ while plot (b) has $T/k_0=0.0798$. The characteristic width of the peaks in these plots is now determined by the temperature scale, so that $\Delta x |_\text{peak} \sim \pi r_+$. This is the expected statement that the temperature serves as the short distance cutoff on the disorder distribution at the horizon.
• Figure 4: Disordered horizon - Ricci scalar. Plot of the Ricci scalar of the induced metric on a spatial cross section of the horizon. The parameters used are the same as in Fig. \ref{['figs:3']}. Because the metric depends on the square of the scalar field, the metric functions oscillate twice as quickly and hence the structures appear half the size of those in Fig. \ref{['figs:3']}.