Black holes and the butterfly effect

Shenker, Stephen H.; Stanford, Douglas

doi:10.1007/jhep03(2014)067

Cited by 1,342 publications

(1,998 citation statements)

References 89 publications

(180 reference statements)

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“…At relatively low energies this growth matches the one expected in a theory of gravity [9,10,11], which saturates the chaos bound [12].…”

Section: Introductionsupporting

confidence: 83%

“…However, we cannot trust the computation for such small values of E; at the crossover point cE ∼ 1, the saddle point in the β integral is at β ∼ c ∼ N/J. At such low temperatures, the Schwarzian action discussed in the previous section stops being semiclassical, and our analysis would need to be improved 9 .…”

Section: The Density Of States and The Free Energymentioning

confidence: 94%

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Remarks on the Sachdev-Ye-Kitaev model

2016

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We study a quantum mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of N Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field.The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to SL(2, R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out of time ordered correlator). We expect these features to be universal properties of large N quantum mechanics systems with emergent reparametrization symmetry.This article is largely based on talks given by Kitaev [1], which motivated us to work out the details of the ideas described there.

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“…At relatively low energies this growth matches the one expected in a theory of gravity [9,10,11], which saturates the chaos bound [12].…”

Section: Introductionsupporting

confidence: 83%

Section: The Density Of States and The Free Energymentioning

confidence: 94%

Remarks on the Sachdev-Ye-Kitaev model

2016

Self Cite

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“…More generally, one would like to identify vB with a universally defined nonquasiparticle velocity. It has been noted (54) that, in some nonquasiparticle systems, it is the "butterfly velocity" (55,56) that appears in the diffusivity equation (Eq. 3).…”

Section: Significancementioning

confidence: 99%

Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x

Zhang

Levenson-Falk

Ramshaw

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

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The thermal diffusivity in the ab plane of underdoped YBCO crystals is measured by means of a local optical technique in the temperature range of 25-300 K. The phase delay between a point heat source and a set of detection points around it allows for high-resolution measurement of the thermal diffusivity and its in-plane anisotropy. Although the magnitude of the diffusivity may suggest that it originates from phonons, its anisotropy is comparable with reported values of the electrical resistivity anisotropy. Furthermore, the anisotropy drops sharply below the charge order transition, again similar to the electrical resistivity anisotropy. Both of these observations suggest that the thermal diffusivity has pronounced electronic as well as phononic character. At the same time, the small electrical and thermal conductivities at high temperatures imply that neither well-defined electron nor phonon quasiparticles are present in this material. We interpret our results through a strongly interacting incoherent electron-phonon "soup" picture characterized by a diffusion constant D ∼ v 2 B τ , where v B is the soup velocity, and scattering of both electrons and phonons saturates a quantum thermal relaxation time τ ∼ /k B T.thermal diffusivity | bad metals | electron-phonon T he standard paradigm for transport in metals relies on the existence of quasiparticles. Electronic quasiparticles conduct electricity and heat. Phonon quasiparticles, the collective excitations of the elastic solid (here, we discuss acoustic phonons) also conduct heat. Transport coefficients, such as electrical and thermal conductivities, can then be calculated using, for example, Boltzmann equations (1). However, such an approach fails when the quasiparticle mean free paths become comparable with the quasiparticle wavelength. For electrons, it is the Fermi wavelength (2-4), whereas for phonons, it is the larger of the interatomic distance or minimum excited phonon wavelength (5, 6). Understanding transport in nonquasiparticle regimes requires a new framework and has become a subject of intense theoretical effort in recent years, triggering an urgent need for experimental results that can shed light on such regimes. In particular, in ref. 7, the diffusivity was singled out as a key observable for incoherent nonquasiparticle transport, possibly subject to fundamental quantum mechanical bounds.In this work, we report high-resolution measurements of the thermal diffusivity of single-crystal underdoped YBCO6.60 (an ortho-II YBa2Cu3O6.60) and YBCO6.75 (an ortho-III YBa2Cu3O6.75). We are particularly interested in the anisotropy of the thermal diffusivity as measured along the principal axes a and b (which is the chain direction) in the temperature range 25-300 K. We use a noncontact optical microscope (see Fig. S1) to perform local thermal transport measurements on the scale of ∼10 µm, hence avoiding inhomogeneities, particularly twinning and grain boundary effects. Our principal experimental results are as follows. (i) The measured thermal diffusivity is...

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“…For more details, we refer to [15,19,[37][38][39][40][41][42][43]. The butterfly effect as chaotic behaviour refers to the exponential growth of a small perturbation to a quantum system.…”

Section: Butterfly Velocitymentioning

confidence: 99%

Diffusion and butterfly velocity at finite density

Niu

Kim

2017

J. High Energ. Phys.

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Abstract: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 − v 2 B /k B T where C − = 1/2 or 1 so it is universal independently of β and µ. However, D c ∼ C + v 2 B /k B T where C + = 1 or β 2 /16π 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.

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Black holes and the butterfly effect

Cited by 1,342 publications

References 89 publications

Remarks on the Sachdev-Ye-Kitaev model

Remarks on the Sachdev-Ye-Kitaev model

Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x

Diffusion and butterfly velocity at finite density

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Black holes and the butterfly effect

Cited by 1,342 publications

References 89 publications

Remarks on the Sachdev-Ye-Kitaev model

Remarks on the Sachdev-Ye-Kitaev model

Anomalous thermal diffusivity in underdoped YBa 2 Cu 3 O 6+x

Diffusion and butterfly velocity at finite density

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Anomalous thermal diffusivity in underdoped YBa ₂ Cu ₃ O _6+x