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Many-body chaos and energy dynamics in holography

Blake, Mike, Davison, Richard A., Grozdanov, Sašo, Liu, Hong·2018·Journal of High Energy Physics
DOI 10.1007/jhep10(2018)035
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Abstract

A bstract: Recent developments have indicated that in addition to out-of-time ordered correlation functions (OTOCs), quantum chaos also has a sharp manifestation in the thermal energy density two-point functions, at least for maximally chaotic systems. The manifestation, referred to as pole-skipping, concerns the analytic behaviour of energy density two-point functions around a special point ω = iλ , k = iλ/v B in the complex frequency and momentum plane. Here λ and v B are the Lyapunov exponent and butterfly velocity characterising quantum chaos. In this paper we provide an argument that the phenomenon of pole-skipping is universal for general finite temperature systems dual to Einstein gravity coupled to matter. In doing so we uncover a surprising universal feature of the linearised Einstein equations around a static black hole geometry. We also study analytically a holographic axion model where all of the features of our general argument as well as the pole-skipping phenomenon can be verified in detail.

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Figure 2 These plots show the dispersion relation ω ( k ) of the hydrodynamic pole in (3.7) as a function of imaginary k for the choices m/T = 1 / 100 and m/T = 100. The blue lines are the hydrodynamic approximations (3.36) (left panel) and (3.37) (right panel) to the small k hydrodynamic behaviour. The black dots correspond to the exact dispersion relation extracted from our numerics. Despite the qualitatively different small k behaviour, in all cases we find this dispersion relation passes through the special point (1.3) such that ω ( ik 0 ) = iλ for k 0 = λ/v B .
Figure 2 These plots show the dispersion relation ω ( k ) of the hydrodynamic pole in (3.7) as a function of imaginary k for the choices m/T = 1 / 100 and m/T = 100. The blue lines are the hydrodynamic approximations (3.36) (left panel) and (3.37) (right panel) to the small k hydrodynamic behaviour. The black dots correspond to the exact dispersion relation extracted from our numerics. Despite the qualitatively different small k behaviour, in all cases we find this dispersion relation passes through the special point (1.3) such that ω ( ik 0 ) = iλ for k 0 = λ/v B .CC-BY · From the PDF
Figure 3 (a) The red line plots the butterfly velocity v B = λ/k 0 as determined from the analytic formula (3.3). The black dots correspond to λ/k 1 where k 1 is extracted as the wavevector for which the numerical dispersion relation for the hydrodynamic pole satisfies ω ( ik 1 ) = iλ . If the pole passes through (1.3) then we should have λ/k 1 = v B which indeed holds for all m/T . (b) The red line plots the analytic formula (3.28) for the slope of the line of poles in the energy density correlator as it passes through the point (1.3). The dots correspond to the values of the slope extracted from the numerical calculations of the dispersion relation. Note that m/T →∞ corresponds to m = √ 6 r 0 , which is the upper limit shown on each plot.
Figure 3 (a) The red line plots the butterfly velocity v B = λ/k 0 as determined from the analytic formula (3.3). The black dots correspond to λ/k 1 where k 1 is extracted as the wavevector for which the numerical dispersion relation for the hydrodynamic pole satisfies ω ( ik 1 ) = iλ . If the pole passes through (1.3) then we should have λ/k 1 = v B which indeed holds for all m/T . (b) The red line plots the analytic formula (3.28) for the slope of the line of poles in the energy density correlator as it passes through the point (1.3). The dots correspond to the values of the slope extracted from the numerical calculations of the dispersion relation. Note that m/T →∞ corresponds to m = √ 6 r 0 , which is the upper limit shown on each plot.CC-BY · From the PDF

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Citations

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Metadata

Authors
Blake, M., Davison, R. A., Grozdanov, S., & Liu, H.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2018-10
Volume
2018
Issue
10
ISSN
1029-8479
DOI
10.1007/jhep10(2018)035
Cited by
158
References
2
Language
en
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