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arXiv:1608.03286 · Journal of High Energy Physics

Charge diffusion and the butterfly effect in striped holographic matter

Andrew Lucas, Julia Steinberg

Published Oct 1, 2016• 75 citations
arXivHTMLPDF

Abstract

Recently, it has been proposed that the butterfly velocity — a speed at which quantum information propagates — may provide a fundamental bound on diffusion constants in dirty incoherent metals. We analytically compute the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength “hydrodynamic” disorder in a single spatial direction. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.

DOI
10.1007/jhep10(2016)143

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Recently, it has been proposed that the butterfly velocity — a speed at which quantum information propagates — may provide a fundamental bound on diffusion constants in dirty incoherent metals. We analytically compute the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength “hydrodynamic” disorder in a single spatial direction. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.
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We study a relation between the thermal diffusivity (D T ) and two quantum chaotic properties, Lyapunov time (τ L ) and butterfly velocity (v B ) in strongly correlated systems by using a holographic method. Recently, it was shown that Ei:=DT,i/(vB,i2τL)(i=x,y) {\mathbb{E}}_i:={D}_{T,i}/\left({v}_{{}^{B,i}}^2{\tau}_L\right)\left(i=x,y\right) Ei​:=DT,i​/(vB,i2​τL​)(i=x,y) is universal in the sense that it is determined only by some scaling exponents of the IR metric in the low temperature limit regardless of the matter fields and ultraviolet data. Inspired by this observation, by analyzing the anisotropic IR scaling geometry carefully, we find the concrete expressions for Ei {\mathbb{E}}_i Ei​ in terms of the critical dynamical exponents z i in each direction, Ei=zi/2(zi−1) {\mathbb{E}}_i={z}_i/2\left({z}_i-1\right) Ei​=zi​/2(zi​−1) . Furthermore, we find the lower bound of Ei {\mathbb{E}}_i Ei​ is always 1/2, which is not affected by anisotropy, contrary to the η/s case. However, there may be an upper bound determined by given fixed anisotropy.
The speed of information propagation is finite in quantum systems with local interactions. In many such systems, local operators spread ballistically in time and can be characterized by a ``butterfly velocity", which can be measured via out-of-time-ordered correlation functions. In general, the butterfly velocity can depend asymmetrically on the direction of information propagation. In this work, we construct a family of simple 2-local Hamiltonians for understanding the asymmetric hydrodynamics of operator spreading. Our models live on a one dimensional lattice and exhibit asymmetric butterfly velocities between the left and right spatial directions. This asymmetry is transparently understood in a free (non-interacting) limit of our model Hamiltonians, where the butterfly speed can be understood in terms of quasiparticle velocities.
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 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 $\ensuremath{\tau}\ensuremath{\sim}\ensuremath{\hbar}/({k}_{B}T)$ independently of the mechanism of momentum relaxation.
Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.