Upper bound of the charge diffusion constant in holography

Abstract

Abstract We investigate the upper bound of charge diffusion constant in holography. For this purpose, we apply the conjectured upper bound proposal related to the equilibration scales (ωeq, keq) to the Einstein-Maxwell-Axion model. (ωeq, keq) is defined as the collision point between the diffusive hydrodynamic mode and the first non-hydrodynamic mode, giving rise to the upper bound of the diffusion constant D at low temperature T as $$ D={\omega}_{\mathrm{eq}}/{k}_{\mathrm{eq}}^2 $$ D = ω eq / k eq 2 . We show that the upper bound proposal also works for the charge diffusion and (ωeq, keq), at low T, is determined by D and the scaling dimension ∆(0) of an infra-red operator as $$ \left({\omega}_{\mathrm{eq}},{k}_{\mathrm{eq}}^2\right)=\left(2\pi T\Delta (0),{\omega}_{\mathrm{eq}}/D\right) $$ ω eq k eq 2 = 2 πT Δ 0 ω eq / D , as for other diffusion constants. However, for the charge diffusion, we find that the collision occurs at real keq, while it is complex for other diffusions. In order to examine the universality of the conjectured upper bound, we also introduce a higher derivative coupling to the Einstein-Maxwell-Axion model. This coupling is particularly interesting since it leads to the violation of the lower bound of the charge diffusion constant so the correction may also have effects on the upper bound of the charge diffusion. We find that the higher derivative coupling does not affect the upper bound so that the conjectured upper bound would not be easily violated.