Universal diffusion in incoherent black holes

We study charge and energy diffusion in simple holographic theories with broken translational symmetry. We find that when the effects of momentum relaxation are very strong the diffusion constants take universal values $D_c\sim D_e\sim \hslash v_B^2/(k_{B}T)$. Here $v_B$ is the velocity of the butterfly effect and the coefficients of proportionality depend only on the scaling exponents of the infra-red fixed point. Our results suggest that diffusion in these incoherent black holes is controlled by $\tau \sim \hslash /(k_{B}T)$ independently of the mechanism of momentum relaxation.

Figure 1

Diffusion constants of the neutral axion model. The diffusion constants themselves (top) are sensitive to the ratio $k/T$. However, upon dividing by the butterfly velocity $v_B^2$ (bottom) the universality is evident. In particular, note that the energy diffusion constant (dashed) becomes universal at large $k/T$. The diffusion constants cross at the self-dual point [28] $k{L}^2=\sqrt{2}{r}_0$.