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Diffusion and Butterfly Velocity at Finite Density

Keun-Young Kim, Chao Niu

Abstract

We study diffusion and butterfly velocity ($v_B$) 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 $D_\pm$ describing the coupled diffusion of charge and energy. By computing $D_\pm$ exactly, we find that in the incoherent regime ($β/T \gg 1,\ β/μ\gg 1$) $D_+$ 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 $β$ and $μ$. However, $D_c \sim C_+ \hbar v_B^2 / k_B T$ where $C_+ = 1$ or $ β^2/16π^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.

Diffusion and Butterfly Velocity at Finite Density

Abstract

We study diffusion and butterfly velocity () in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter () at finite density or chemical potential (). Axion-dilaton model is particularly interesting since it shows linear--resistivity, which may have something to do with the universal bound of diffusion. At finite density, there are two diffusion constants describing the coupled diffusion of charge and energy. By computing exactly, we find that in the incoherent regime () is identified with the charge diffusion constant () and is identified with the energy diffusion constant (). In the coherent regime, at very small density, are `maximally' mixed in the sense that is identified with , which is opposite to the case in the incoherent regime. In the incoherent regime where or 1 so it is universal independently of and . However, where or so, in general, 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.

Paper Structure

This paper contains 10 sections, 69 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Diffusion constants
  4. Butterfly velocity
  5. Linear axion model
  6. Axion-dilaton model
  7. Thermodynamics and transport coefficients
  8. Zero density
  9. Finite density
  10. Conclusion

Figures (6)

  • Figure 1: Diffusion constants of the linear axion model at finite density. $\mu/T=0.1, 1, 5$ from left to right. The blue curve is for $D_+$ and the red curve is for $D_-$. The green curve displays $v_B^2$. $2\pi D_{\pm}/v_B^2$ saturates the universal value in the incoherent regime: $\beta/T \gg 1$ and $\beta/\mu \gg 1$.
  • Figure 2: Diffusion constants of the linear axion model with/without a mixing term. $\mu/T=0.1, 1, 5$ from left to right. The blue curve is for $D_c$ and the red curve is for $D_e$. The triangles display the results without the mixing term $\mathcal{M}$ in \ref{['c1c22']}. For comparison, we also display the results with the mixing term, the solid curves in Fig. \ref{['fig1']}.
  • Figure 3: The left panel: two values of ${\tilde{Q}}$ for given $\mu/T=0.1$(a) and $\mu/T=5$(b). The red curve is for positive ${\tilde{Q}}$ and the blue curve is for negative ${\tilde{Q}}$. The right panel: the difference of the grand potential ($\delta {G} = {G}({\tilde{Q}}>0) - {G}({\tilde{Q}}<0)$ is shown. The positive ${\tilde{Q}}$ solution is always thermodynamically preferred. The green region represent two branches in \ref{['branches']}.
  • Figure 4: Diffusion constants of the axion-dilaton model at zero density for ${\tilde{Q}}=0$ (a), for ${\tilde{Q}} = -1 + \frac{{\tilde{\beta}}}{\sqrt{2}}$ (b), and for the ground state (c). The blue curve is for $D_c$ and the red curve is for $D_e$. The green curve displays $v_B^2$.
  • Figure 5: Diffusion constants of the axion-dilaton model at finite density. $\mu/T=0.1, 1, 5$ from left to right. The blue curve is for $D_+$ and the red curve is for $D_-$. The green curve displays $v_B^2$. $2\pi D_+$ and $2\pi T D_{-}/v_B^2$ saturates the universal value in the incoherent regime: $\beta/T \gg 1$ and $\beta/\mu \gg 1$.
  • ...and 1 more figures