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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 timescal

Author

Jeong, Hyun-Sik; Kim, Keun-Young; Sun, Ya-Wen

Published

2021

Language

EN