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arXiv:1604.01754 · Physical review. D/Physical review. D.

Universal diffusion in incoherent black holes

Mike Blake

Published Oct 31, 2016• 123 citations
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Abstract

We study charge and energy diffusion in simple holographic theories with broken translational symmetry. We find that when the effects of momentum relaxation are very strong the diffusion constants take universal values ${D}_{c}\ensuremath{\sim}{D}_{e}\ensuremath{\sim}\ensuremath{\hbar}{v}_{B}^{2}/({k}_{B}T)$. Here ${v}_{B}$ is the velocity of the butterfly effect and the coefficients of proportionality depend only on the scaling exponents of the infra-red fixed point. Our results suggest that diffusion in these incoherent black holes is controlled by $\ensuremath{\tau}\ensuremath{\sim}\ensuremath{\hbar}/({k}_{B}T)$ independently of the mechanism of momentum relaxation.

DOI
10.1103/physrevd.94.086014

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We study charge and energy diffusion in simple holographic theories with broken translational symmetry. We find that when the effects of momentum relaxation are very strong the diffusion constants take universal values Dc\ensuremath∼De\ensuremath∼\ensuremathℏvB2/(kBT){D}_{c}\ensuremath{\sim}{D}_{e}\ensuremath{\sim}\ensuremath{\hbar}{v}_{B}^{2}/({k}_{B}T)Dc​\ensuremath∼De​\ensuremath∼\ensuremathℏvB2​/(kB​T). Here vB{v}_{B}vB​ is the velocity of the butterfly effect and the coefficients of proportionality depend only on the scaling exponents of the infra-red fixed point. Our results suggest that diffusion in these incoherent black holes is controlled by \ensuremathτ\ensuremath∼\ensuremathℏ/(kBT)\ensuremath{\tau}\ensuremath{\sim}\ensuremath{\hbar}/({k}_{B}T)\ensuremathτ\ensuremath∼\ensuremathℏ/(kB​T) independently of the mechanism of momentum relaxation.
The dissipative dynamics of strongly interacting systems are often characterized by the timescale set by the inverse temperature ${\ensuremath{\tau}}_{P}\ensuremath{\sim}\ensuremath{\hbar}/({k}_{B}T)$. We show that near a class of strongly interacting quantum critical points that arise in the infrared limit of translationally invariant holographic theories, there is a collective excitation (a quasinormal mode of the dual black hole spacetime) whose lifetime ${\ensuremath{\tau}}_{\mathrm{eq}}$ is parametrically longer than ${\ensuremath{\tau}}_{P}$: ${\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\gg}{T}^{\ensuremath{-}1}$. The lifetime is enhanced due to its dependence on a dangerously irrelevant coupling that breaks the particle-hole symmetry and the invariance under Lorentz boosts of the quantum critical point. The thermal diffusivity (in units of the butterfly velocity) is anomalously large near the quantum critical point and is governed by ${\ensuremath{\tau}}_{\mathrm{eq}}$ rather than ${\ensuremath{\tau}}_{P}$. We conjecture that there exists a long-lived, propagating collective mode with velocity vs{v}_{s}vs​, and in this case the relation $D={v}_{s}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}$ holds exactly in the limit $T{\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\gg}1$. While scale invariance is broken, a generalized scaling theory still holds provided that the dependence of observables on the dangerously irrelevant coupling is incorporated. Our work further underlines the connection between dangerously irrelevant deformations and slow equilibration.
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In this Letter, we uncover a universal relaxation mechanism of pinned density waves, combining gauge-gravity duality and effective field theory techniques. Upon breaking translations spontaneously, new gapless collective modes emerge, the Nambu-Goldstone bosons of broken translations. When translations are also weakly broken (e.g., by disorder or lattice effects), these phonons are pinned with a mass m and damped at a rate Ω, which we explicitly compute. This contribution to Ω is distinct from that of topological defects. We show that Ω≃Gm^{2}Ξ, where G is the shear modulus and Ξ is related to a diffusivity of the purely spontaneous state. This result follows from the smallness of the bulk and shear moduli, as would be the case in a phase with fluctuating translational order. At low temperatures, the collective modes relax quickly into the heat current, so that late time transport is dominated by the thermal diffusivity. In this regime, the resistivity in our model is linear in temperature and the ac conductivity displays a significant rearranging of the degrees of freedom, as spectral weight is shifted from an off-axis, pinning peak to a Drude-like peak. These results could shed light on transport properties in cuprate high T_{c} superconductors, where quantum critical behavior and translational order occur over large parts of the phase diagram and transport shows qualitatively similar features.
In a clean quantum critical metal, and in the absence of umklapp, most d.c. conductivities are formally infinite due to momentum conservation. However, there is a particular combination of the charge and heat currents which has a finite, universal conductivity. In this paper, we describe the physics of this conductivity σ Q in quantum critical metals obtained by charge doping a strongly interacting conformal field theory. We show that it satisfies an Einstein relation and controls the diffusivity of a conserved charge in the metal. We compute σ Q in a class of theories with holographic gravitational duals. Finally, we show how the temperature scaling of σ Q depends on certain critical exponents characterizing the quantum critical metal. The holographic results are found to be reproduced by the scaling analysis, with the charge density operator becoming marginal in the emergent low energy quantum critical theory.
We study entanglement in thermofield double states of strongly coupled conformal field theories by analyzing two-sided Reissner-Nordstr\"om solutions in anti--de Sitter space. The central object of study is the mutual information between a pair of regions, one on each asymptotic boundary of the black hole. For large regions the mutual information is positive and for small ones it vanishes; we compute the critical length scale, which goes to infinity for extremal black holes, of the transition. We also generalize the butterfly effect of Shenker and Stanford [J. High Energy Phys. 03 (2014) 067] to a wide class of charged black holes, showing that mutual information is disrupted upon perturbing the system and waiting for a time of order $\mathrm{log}E/\ensuremath{\delta}E$ in units of the temperature. We conjecture that the parametric form of this time scale is universal.