Diffusion and Butterfly Velocity at Finite Density

TL;DR

This paper investigates diffusion constants and butterfly velocity in holographic models with momentum relaxation at finite density, revealing universal behaviors and regimes where charge and energy diffusion are coupled or separated.

Contribution

It provides exact calculations of coupled charge and energy diffusion constants at finite density, identifying regimes where these are maximally mixed or separated, and explores their relation to butterfly velocity.

Findings

Incoherent regime: $D_+$ and $D_-$ correspond to charge and energy diffusion.

Universal relation: $D_e o rac{1}{2} rac{ar{ u} v_B^2}{T}$ independent of parameters.

Charge diffusion constant may not saturate the lower bound set by butterfly velocity.

Abstract

We study diffusion and butterfly velocity ($v_B$) in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter ($\beta$) at finite density or chemical potential ($\mu$). Axion-dilaton model is particularly interesting since it shows linear-$T$-resistivity, which may have something to do with the universal bound of diffusion. At finite density, there are two diffusion constants $D_\pm$ describing the coupled diffusion of charge and energy. By computing $D_\pm$ exactly, we find that in the incoherent regime ($\beta/T \gg 1,\ \beta/\mu \gg 1$) $D_+$ is identified with the charge diffusion constant ($D_c$) and $D_-$ is identified with the energy diffusion constant ($D_e$). In the coherent regime, at very small density, $D_\pm$ are `maximally' mixed in the sense that $D_+(D_-)$ is identified with $D_e(D_c)$, which is opposite to the case in the incoherent regime. In the incoherent regime $D_e \sim C_- \hbar v_B^2 / k_B T$ where $C_- = 1/2$ or 1 so it is universal independently of $\beta$ and $\mu$. However, $D_c \sim C_+ \hbar v_B^2 / k_B T$ where $C_+ = 1$ or $ \beta^2/16\pi^2 T^2$ so, in general, $C_+$ may not saturate to the lower bound in the incoherent regime, which suggests that the characteristic velocity for charge diffusion may not be the butterfly velocity. We find that the finite density does not affect the diffusion property at zero density in the incoherent regime.