The breakdown of magneto-hydrodynamics near AdS2 fixed point and energy diffusion bound

Abstract

Abstract We investigate the breakdown of magneto-hydrodynamics at low temperature (T) with black holes whose extremal geometry is AdS2×R2. The breakdown is identified by the equilibration scales (ωeq, keq) defined as the collision point between the diffusive hydrodynamic mode and the longest-lived non-hydrodynamic mode. We show (ωeq, keq) at low T is determined by the diffusion constant D and the scaling dimension ∆(0) of an infra-red operator: ωeq = 2πT∆(0),$$ {k}_{\mathrm{eq}}^2 $$ k eq 2 = ωeq/D, where ∆(0) = 1 in the presence of magnetic fields. For the purpose of comparison, we have analytically shown ∆(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 $$ D\le {\omega}_{\mathrm{eq}}/{k}_{\mathrm{eq}}^2:= {v}_{\mathrm{eq}}^2{\tau}_{\mathrm{eq}} $$ D ≤ ω eq / k eq 2 ≔ v eq 2 τ eq where veq := ωeq/keq and τeq := $$ {\omega}_{\mathrm{eq}}^{-1} $$ ω 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.