Universal Charge Diffusion and the Butterfly Effect

TL;DR

The paper investigates charge diffusion in holographic theories with particle-hole symmetry and IR scaling geometries characterized by exponents $z$, \theta, and a charge-dimension anomaly $\Phi$. By computing the butterfly velocity $v_B$ from horizon shock-wave analysis and the diffusion constant $D_c$ via the membrane paradigm, it shows a universal IR regime where $D_c = \frac{d_\theta}{\Delta_\chi} \frac{v_B^2}{2\pi T}$, linking transport to quantum chaos. The results demonstrate that, in the IR, diffusion is governed by horizon data and can be bounded (up to order-one coefficients) by chaos parameters, while UV data controls diffusion in non-universal regimes. The Appendix extends the analysis to momentum diffusion, supporting the broader claim that diffusion constants in holographic theories are intrinsically tied to the butterfly effect, with potential extensions to anisotropic and momentum-relaxing systems.

Abstract

We study charge diffusion in holographic scaling theories with a particle-hole symmetry. We show that these theories have a universal regime in which the diffusion constant is given by $D_c = C v_B^2/ (2 πT)$ where $v_B$ is the velocity of the butterfly effect. The constant of proportionality, $C$, depends only on the scaling exponents of the infra-red theory. Our results suggest an unexpected connection between transport at strong coupling and quantum chaos.