The breakdown of magneto-hydrodynamics near AdS$_2$ fixed point and energy diffusion bound

TL;DR

This paper explores the conditions under which magneto-hydrodynamics breaks down near an AdS$_2$ fixed point at low temperatures, linking the breakdown scales to diffusion constants and IR operator dimensions, and supporting a universal diffusion bound.

Contribution

It analytically determines the equilibration scales and diffusion bounds near AdS$_2$ fixed points, including the IR operator dimension in magnetic and axion models, extending previous numerical results.

Findings

Equilibration scales are determined by diffusion constant and IR operator dimension.

The IR operator dimension is analytically shown to be 1 with magnetic fields and 2 in axion models.

Supports the universal upper bound of energy diffusion related to hydrodynamic breakdown.

Abstract

We investigate the breakdown of magneto-hydrodynamics at low temperature ($T$) with black holes whose extremal geometry is AdS$_2\times$R$^2$. The breakdown is identified by the equilibration scales ($\omega_{\text{eq}}, k_{\text{eq}}$) defined as the collision point between the diffusive hydrodynamic mode and the longest-lived non-hydrodynamic mode. We show ($\omega_{\text{eq}}, k_{\text{eq}}$) at low $T$ is determined by the diffusion constant $D$ and the scaling dimension $\Delta(0)$ of an infra-red operator: $\omega_{\text{eq}} = 2\pi T \Delta(0), \, k_{\text{eq}}^2 = \omega_{\text{eq}}/D$, where $\Delta(0)=1$ in the presence of magnetic fields. For the purpose of comparison, we have analytically shown $\Delta(0)=2$ for the axion model independent of the translational symmetry breaking pattern (explicit or spontaneous), which is complementary to previous numerical results. Our results support the conjectured universal upper bound of the energy diffusion $D \,\le\, \omega_{\text{eq}}/k_{\text{eq}}^2 \,:=\, v_{\text{eq}}^2 \, \tau_{\text{eq}}$ where $v_{\text{eq}}:= \omega_{\text{eq}}/k_{\text{eq}}$ and $\tau_{\text{eq}}:=\omega_{\text{eq}}^{-1}$ are the velocity and the timescale associated to equilibration, implying that the breakdown of hydrodynamics sets the upper bound of the diffusion constant $D$ at low $T$.