Charge diffusion and the butterfly effect in striped holographic matter

TL;DR

Charged diffusion in strongly interacting, translation-symmetry-broken holographic matter is studied by constructing striped black hole backgrounds via the fluid-gravity expansion. The authors compute the diffusion constant $D$ and butterfly velocity $v_{\textsc{b}}$ in a charge-neutral setting, deriving explicit horizon-based expressions: $\sigma = \dfrac{1}{\mathbb{E}[ \dfrac{e^2}{Z} b^{-(d-2)/2} ]}$ and $\chi = \mathbb{E}\left[\left(\int_0^{r_+} \! dr \frac{e^2 \tilde{a}}{Z b^{d/2}}\right)^{-1}\right]$, with $D = \sigma/\chi$. They connect $v_{\textsc{b}}$ to the shockwave geometry, obtaining $v_{\textsc{b}}$ from a horizon perturbation and showing, in IR scaling geometries, $D$ satisfies $2 \pi T D \le C v_{\textsc{b}}^2$, with equality only in homogeneous backgrounds. The main result is that the butterfly velocity does not generally set a sharp lower bound for $D$ in striped holographic matter, challenging the universality of the diffusion bound in strongly coupled, disordered metals.

Abstract

Recently, it has been proposed that the butterfly velocity - a speed at which quantum information propagates - may provide a fundamental bound on diffusion constants in dirty incoherent metals. We analytically compute the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength "hydrodynamic" disorder in a single spatial direction. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.

Figures (1)

• Figure 1: (a) The global Kruskal geometry. The smooth black lines on the left/right side of the diagram denote the "AdS" boundaries of the left/right field theories. The rough red lines at the top and bottom denote the singularities inside of a black hole and a white hole respectively. The dashed purple lines denote event horizons at $U=0$ and $V=0$. Blue arrows denote the direction as well as the standard "bulk radial" coordinate $r$. Eddington-Finkelstein coordinates only cover the region $V\ge 0$ in this diagram. The properties of this black hole vary smoothly in the $x$ direction, which runs out of the page. (b) The effect of the "geometric shockwave" is a shift $h(x)$ between the horizons as one crosses $U=0$.