Disordered horizons: Holography of randomly disordered fixed points

TL;DR

This work investigates how marginally relevant random disorder deforms a UV CFT with a holographic gravity dual. By introducing a Gaussian random boundary source for a neutral scalar and solving the bulk Einstein equations perturbatively, the authors identify an IR fixed point with Lifshitz scaling, characterized by a dynamical exponent zbar that exceeds unity and grows with disorder. Analytically, zbar is obtained via resummed perturbation theory, and numerically confirmed, with explicit results such as zbar = 1 + 1/2 Vbar^2 + 1/2 ln(2) Vbar^4 + O(Vbar^6) in D=3; general D expressions are also derived. The findings establish a holographic realization of finite-randomness IR fixed points and suggest rich transport and localization physics in strongly interacting systems under disorder.

Abstract

We deform conformal field theories with classical gravity duals by marginally relevant random disorder. We show that the disorder generates a flow to IR fixed points with a finite amount of disorder. The randomly disordered fixed points are characterized by a dynamical critical exponent $z>1$ that we obtain both analytically (via resummed perturbation theory) and numerically (via a full simulation of the disorder). The IR dynamical critical exponent increases with the magnitude of disorder, probably tending to $z \to \infty$ in the limit of infinite disorder.

Figures (3)

• Figure 1: Disordered horizons: Plot (a) shows the scalar source $\Phi_1$ as a function of $x\,k_0$ at the boundary and the scalar field $\Phi$ at $z=8$, which is well within the IR Lifshitz scaling regime. Plot (b) is a density plot of $\Phi$, now in a $D=4$ spacetime, for $\ell = 1$, as a function of boundary directions $x\,k_0$ and $y\,k_0$. In both cases, we used $N=10$, $k_0=5$ and $\bar{V}=0.1$.
• Figure 2: Emergence of an IR dynamical scaling exponent. Plot of ${\overline z}(z)$, the logarithmic derivative (\ref{['eq:logderiv']}) of $\langle g_{tt}\rangle_R$, for several values of $\bar{V}$. In the UV, ${\overline z} = 1$ for all curves. The constant values at large $z$ indicate an emergent Lifshitz scaling with ${\overline z}$ increasing with $\bar{V}$. These plots have $N=50$ and $k_0=5$.
• Figure 3: Large disorder leads to large ${\overline z}$: Dynamical critical exponent ${\overline z}$ as a function of $\bar{V}$. Plot (a) is for D=3 and plot (b) is for D=4. Numerical results are shown with statistical error bars while the dashed line is the perturbative expansion in $\bar{V}$. In $D=3$ we go well beyond the perturbative regime, while in $D=4$ we obtain a check of our analytic result.