Diffusion and Butterfly Velocity at Finite Density

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 $D_e\sim C_{-}\hslash v_B^2/k_{B}T$ where $C_{-}=1/2$ or 1 so it is universal independently of $\beta$ and $\mu$. However, $D_c\sim C_{+}\hslash v_B^2/k_{B}T$ 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.

24 pages, 6 figures, v2 minor edits and references added

References in corpus (8)

• Theory of universal incoherent metallic transport
• Thermoelectric transport in disordered metals without quasiparticles: the SYK models and holography
• Holographic quantum matter
• Slow scrambling in disordered quantum systems
• Momentum dissipation and effective theories of coherent and incoherent transport
• Quantum chaos on a critical Fermi surface
• Coherent/incoherent metal transition in a holographic model
• Higher derivative corrections to incoherent metallic transport in holography

Cited by in corpus (5)

• Boomerang RG flows with intermediate conformal invariance
• Collective diffusion and quantum chaos in holography
• Diffusion in inhomogeneous media
• Constraints on hydrodynamics from many-body quantum chaos
• Diffusivities bounds in the presence of Weyl corrections