Diffusion and butterfly velocity at finite density

• Published in: The journal of high energy physics 2017-06, Vol.2017 (6), p.1-24, Article 30
• Main Authors: Niu, Chao, Kim, Keun-Young
• Format: Article
• Language: English
• Subjects: Chemical potential; Classical and Quantum Gravitation; Density; Elementary Particles; Gauge-gravity correspondence; High energy physics; Holography and condensed matter physics (AdS/CMT); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Velocity
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP06(2017)030

A bstract 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 ± describing the coupled diffusion of charge and energy. By computing D ± exactly, we find that in the incoherent regime ( β/T ≫ 1 , β/μ ≫ 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 ± 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  ∼  C − ℏv B 2 / k B T where C − = 1 / 2 or 1 so it is universal independently of β and μ . However, D c ∼ C + ℏ v B 2 / k B T where C + = 1 or β 2 / 16 π 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.