Diffusion and butterfly velocity at finite density

Abstract

We study diffusion and butterfly velocity (vB) in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter (β) at finite density or chemical potential (μ). 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± describing the coupled diffusion of charge and energy. By computing D± exactly, we find that in the incoherent regime (β/T ≫ 1, β/μ ≫ 1) D+ is identified with the charge diffusion constant (Dc) and D− is identified with the energy diffusion constant (De). In the coherent regime, at very small density, D± are ‘maximally’ mixed in the sense that D+(D−) is identified with De(Dc), which is opposite to the case in the incoherent regime. In the incoherent regime De ∼ C−ℏvB2/kBT where C− = 1/2 or 1 so it is universal independently of β and μ. However, {D}_csim {C}_{+}hslash {v}_{{}^B}^2/{k}_BT where C+ = 1 or β2/16π2T2 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.