Abstract
When the Lyapunov exponent λL in a quantum chaotic system saturates the bound λL ≤ 2πk B T , it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator transitions (MIT), in which the derivatives of the butterfly velocity with respect to system parameters characterizes quantum critical points (QCP) with local extremes in zero temperature limit. We also point out that this proposal can be tested by experiments in the light of recent progress on the measurement of out-of-time-order correlation function (OTOC).
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References
D.A. Roberts and B. Swingle, Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories, Phys. Rev. Lett. 117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J.M. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
J. Polchinski, Chaos in the black hole S-matrix, arXiv:1505.08108 [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94 (2016) 040302 [arXiv:1602.06271] [INSPIRE].
M. Blake, Universal Charge Diffusion and the Butterfly Effect in Holographic Theories, Phys. Rev. Lett. 117 (2016) 091601 [arXiv:1603.08510] [INSPIRE].
M. Blake, Universal Diffusion in Incoherent Black Holes, Phys. Rev. D 94 (2016) 086014 [arXiv:1604.01754] [INSPIRE].
A. Lucas and J. Steinberg, Charge diffusion and the butterfly effect in striped holographic matter, JHEP 10 (2016) 143 [arXiv:1608.03286] [INSPIRE].
A. Kitaev, Hidden correlations in the hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, Stanford University, Stanford, U.S.A., 10 November 2014.
S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].
H. Shen, P. Zhang, R. Fan and H. Zhai, Out-of-Time-Order Correlation at a Quantum Phase Transition, Phys. Rev. B 96 (2017) 054503 [arXiv:1608.02438] [INSPIRE].
R. Fan, P. Zhang, H. Shen and H. Zhai, Out-of-Time-Order Correlation for Many-Body Localization, arXiv:1608.01914 [INSPIRE].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic Entanglement Entropy Close to Quantum Phase Transitions, JHEP 04 (2016) 114 [arXiv:1502.03661] [INSPIRE].
Y. Ling, P. Liu and J.-P. Wu, Characterization of Quantum Phase Transition using Holographic Entanglement Entropy, Phys. Rev. D 93 (2016) 126004 [arXiv:1604.04857] [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].
R.-G. Cai, X.-X. Zeng and H.-Q. Zhang, Influence of inhomogeneities on holographic mutual information and butterfly effect, JHEP 07 (2017) 082 [arXiv:1704.03989] [INSPIRE].
A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].
M. Baggioli and O. Pujolàs, Electron-Phonon Interactions, Metal-Insulator Transitions and Holographic Massive Gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N.Y. Yao et al., Interferometric Approach to Probing Fast Scrambling, arXiv:1607.01801 [INSPIRE].
M. Gärttner, J.G. Bohnet, A. Safavi-Naini, M.L. Wall, J.J. Bollinger and A.M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet, arXiv:1608.08938 [INSPIRE].
J. Li et al., Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator, Phys. Rev. X 7 (2017) 031011 [arXiv:1609.01246] [INSPIRE].
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Ling, Y., Liu, P. & Wu, JP. Holographic butterfly effect at quantum critical points. J. High Energ. Phys. 2017, 25 (2017). https://doi.org/10.1007/JHEP10(2017)025
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DOI: https://doi.org/10.1007/JHEP10(2017)025