Abstract
We use holographic methods to study several chaotic properties of a super Yang-Mills theory at temperature T in the presence of a background magnetic field of constant strength B. The field theory we work on has a renormalization flow between a fixed point in the ultraviolet and another in the infrared, occurring in such a way that the energy at which the crossover takes place is a monotonically increasing function of the dimensionless ratio ℬ/T2. By considering shock waves in the bulk of the dual gravitational theory, and varying ℬ/T2, we study how several chaos-related properties of the system behave while the theory they live in follows the renormalization flow. In particular, we show that the entanglement and butterfly velocities generically increase in the infrared theory, violating the previously suggested upper bounds but never surpassing the speed of light. We also investigate the recent proposal relating the butterfly velocity with diffusion coefficients. We find that electric diffusion constants respect the lower bound proposed by Blake. All our results seem to consistently indicate that the global effect of the magnetic field is to strengthen the internal interaction of the system.
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J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [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].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [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].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, November 10, 2014. Stanford SITP seminars, November 11 and December 18, 2014.
X.-L. Qi and Z. Yang, Butterfly velocity and bulk causal structure, arXiv:1705.01728 [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S. Leichenauer, Disrupting Entanglement of Black Holes, Phys. Rev. D 90 (2014) 046009 [arXiv:1405.7365] [INSPIRE].
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].
Y. Ling, P. Liu and J.-P. Wu, Holographic Butterfly Effect at Quantum Critical Points, JHEP 10 (2017) 025 [arXiv:1610.02669] [INSPIRE].
W.-H. Huang and Y.-H. Du, Butterfly Effect and Holographic Mutual Information under External Field and Spatial Noncommutativity, JHEP 02 (2017) 032 [arXiv:1609.08841] [INSPIRE].
N. Sircar, J. Sonnenschein and W. Tangarife, Extending the scope of holographic mutual information and chaotic behavior, JHEP 05 (2016) 091 [arXiv:1602.07307] [INSPIRE].
M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP 05 (2017) 065 [arXiv:1608.05101] [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].
D. Giataganas, U. Gürsoy and J.F. Pedraza, Strongly-coupled anisotropic gauge theories and holography, arXiv:1708.05691 [INSPIRE].
V. Jahnke, Delocalizing entanglement of anisotropic black branes, JHEP 01 (2018) 102 [arXiv:1708.07243] [INSPIRE].
M.M. Qaemmaqami, Criticality in third order lovelock gravity and butterfly effect, Eur. Phys. J. C 78 (2018) 47 [arXiv:1705.05235] [INSPIRE].
M.M. Qaemmaqami, Butterfly effect in 3D gravity, Phys. Rev. D 96 (2017) 106012 [arXiv:1707.00509] [INSPIRE].
J. de Boer, E. Llabrés, J.F. Pedraza and D. Vegh, Chaotic strings in AdS/CFT, Phys. Rev. Lett. 120 (2018) 201604 [arXiv:1709.01052] [INSPIRE].
K. Murata, Fast scrambling in holographic Einstein-Podolsky-Rosen pair, JHEP 11 (2017) 049 [arXiv:1708.09493] [INSPIRE].
S.-F. Wu, B. Wang, X.-H. Ge and Y. Tian, Holographic RG flow of thermoelectric transport with momentum dissipation, Phys. Rev. D 97 (2018) 066029 [arXiv:1706.00718] [INSPIRE].
W.-H. Huang, Holographic Butterfly Velocities in Brane Geometry and Einstein-Gauss-Bonnet Gravity with Matters, Phys. Rev. D 97 (2018) 066020 [arXiv:1710.05765] [INSPIRE].
M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [arXiv:1805.01470] [INSPIRE].
W.-H. Huang, Butterfly Velocity in Quadratic Gravity, Class. Quant. Grav. 35 (2018) 195004 [arXiv:1804.05527] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
K. Damle and S. Sachdev, Nonzero-temperature transport near quantum critical points, Phys. Rev. B 56 (1997) 8714 [cond-mat/9705206] [INSPIRE].
S. Sachdev, Quantum phase transitions, Cambrigde University Press, (1999).
S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].
A. Lucas and J. Steinberg, Charge diffusion and the butterfly effect in striped holographic matter, JHEP 10 (2016) 143 [arXiv:1608.03286] [INSPIRE].
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
M. Baggioli, B. Goutéraux, E. Kiritsis and W.-J. Li, Higher derivative corrections to incoherent metallic transport in holography, JHEP 03 (2017) 170 [arXiv:1612.05500] [INSPIRE].
K.-Y. Kim and C. Niu, Diffusion and Butterfly Velocity at Finite Density, JHEP 06 (2017) 030 [arXiv:1704.00947] [INSPIRE].
M. Blake, R.A. Davison and S. Sachdev, Thermal diffusivity and chaos in metals without quasiparticles, Phys. Rev. D 96 (2017) 106008 [arXiv:1705.07896] [INSPIRE].
H.-S. Jeong, Y. Ahn, D. Ahn, C. Niu, W.-J. Li and K.-Y. Kim, Thermal diffusivity and butterfly velocity in anisotropic Q-Lattice models, JHEP 01 (2018) 140 [arXiv:1708.08822] [INSPIRE].
E. D’Hoker and P. Kraus, Magnetic Brane Solutions in AdS, JHEP 10 (2009) 088 [arXiv:0908.3875] [INSPIRE].
G. Arciniega, P. Ortega and L. Patiño, Brighter Branes, enhancement of photon production by strong magnetic fields in the gauge/gravity correspondence, JHEP 04 (2014) 192 [arXiv:1307.1153] [INSPIRE].
D. Areán, L.A. Pando Zayas, L. Patiño and M. Villasante, Velocity Statistics in Holographic Fluids: Magnetized quark-gluon Plasma and Superfluid Flow, JHEP 10 (2016) 158 [arXiv:1606.03068] [INSPIRE].
R.P. Martinez-y Romero, L. Patiño and T. Ramirez-Urrutia, Increase of the Energy Necessary to Probe Ultraviolet Theories Due to the Presence of a Strong Magnetic Field, JHEP 11 (2017) 104 [arXiv:1703.03428] [INSPIRE].
M. Ammon, J. Leiber and R.P. Macedo, Phase diagram of 4D field theories with chiral anomaly from holography, JHEP 03 (2016) 164 [arXiv:1601.02125] [INSPIRE].
M. Ammon, M. Kaminski, R. Koirala, J. Leiber and J. Wu, Quasinormal modes of charged magnetic black branes & chiral magnetic transport, JHEP 04 (2017) 067 [arXiv:1701.05565] [INSPIRE].
M. Rahimi and M. Ali-Akbari, Holographic Entanglement Entropy Decomposition in an Anisotropic Gauge Theory, Phys. Rev. D 98 (2018) 026004 [arXiv:1803.01754] [INSPIRE].
M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].
M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area Laws in Quantum Systems: Mutual Information and Correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].
H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].
M. Mezei, On entanglement spreading from holography, JHEP 05 (2017) 064 [arXiv:1612.00082] [INSPIRE].
A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, DC Conductivity of Magnetised Holographic Matter, JHEP 01 (2016) 113 [arXiv:1511.00713] [INSPIRE].
M. Blake, A. Donos and N. Lohitsiri, Magnetothermoelectric Response from Holography, JHEP 08 (2015) 124 [arXiv:1502.03789] [INSPIRE].
K.-Y. Kim, K.K. Kim, Y. Seo and S.-J. Sin, Thermoelectric Conductivities at Finite Magnetic Field and the Nernst Effect, JHEP 07 (2015) 027 [arXiv:1502.05386] [INSPIRE].
S.A. Hartnoll and P. Kovtun, Hall conductivity from dyonic black holes, Phys. Rev. D 76 (2007) 066001 [arXiv:0704.1160] [INSPIRE].
S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter, and in dyonic black holes, Phys. Rev. B 76 (2007)144502 [arXiv:0706.3215] [INSPIRE].
R.A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, JHEP 01 (2015) 039 [arXiv:1411.1062] [INSPIRE].
A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
E. D’Hoker and B. Pourhamzeh, Emergent super-Virasoro on magnetic branes, JHEP 06 (2016) 146 [arXiv:1602.01487] [INSPIRE].
W. Fischler, V. Jahnke and J. Pedraza, in preparation.
A. Ayala et al., Thermomagnetic properties of the strong coupling in the local Nambu-Jona-Lasinio model, Phys. Rev. D 94 (2016) 054019 [arXiv:1603.00833] [INSPIRE].
A. Ayala, C.A. Dominguez, L.A. Hernandez, M. Loewe and R. Zamora, Inverse magnetic catalysis from the properties of the QCD coupling in a magnetic field, Phys. Lett. B 759 (2016)99 [arXiv:1510.09134] [INSPIRE].
A. Ayala et al., Thermomagnetic evolution of the QCD strong coupling, Phys. Rev. D 98 (2018) 031501 [arXiv:1805.08198] [INSPIRE].
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Ávila, D., Jahnke, V. & Patiño, L. Chaos, diffusivity, and spreading of entanglement in magnetic branes, and the strengthening of the internal interaction. J. High Energ. Phys. 2018, 131 (2018). https://doi.org/10.1007/JHEP09(2018)131
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DOI: https://doi.org/10.1007/JHEP09(2018)131