Abstract
The ratio of shear viscosity to entropy density, η/s, is computed in various holographic geometries that break translation invariance (but are isotropic). The shear viscosity does not have a hydrodynamic interpretation in such backgrounds, but does quantify the rate of entropy production due to a strain. Fluctuations of the metric components δg xy are massive about these backgrounds, leading to η/s < 1/(4π) at all finite temperatures (even in Einstein gravity). As the temperature is taken to zero, different behaviors are possible. If translation symmetry breaking is irrelevant in the far IR, then η/s tends to a constant at T = 0. This constant can be parametrically small. If the translation symmetry is broken in the far IR (which nonetheless develops emergent scale invariance), then η/s ∼ T 2ν as T → 0, with ν ≤ 1 in all cases we have considered. While these results violate simple bounds on η/s, we note that they are consistent with a possible bound on the rate of entropy production due to strain.
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References
P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, The Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Holography and hydrodynamics: Diffusion on stretched horizons, JHEP 10 (2003) 064 [hep-th/0309213] [INSPIRE].
A. Adams, L.D. Carr, T. Schäfer, P. Steinberg and J.E. Thomas, Strongly Correlated Quantum Fluids: Ultracold Quantum Gases, Quantum Chromodynamic Plasmas and Holographic Duality, New J. Phys. 14 (2012) 115009 [arXiv:1205.5180] [INSPIRE].
S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].
A. Lucas, Hydrodynamic transport in strongly coupled disordered quantum field theories, New J. Phys. 17 (2015) 113007 [arXiv:1506.02662] [INSPIRE].
S. Grozdanov, A. Lucas, S. Sachdev and K. Schalm, Absence of disorder-driven metal-insulator transitions in simple holographic models, Phys. Rev. Lett. 115 (2015) 221601 [arXiv:1507.00003] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, arXiv:1503.01409 [INSPIRE].
Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].
A. Buchel, R.C. Myers and A. Sinha, Beyond η/s = 1/4π, JHEP 03 (2009) 084 [arXiv:0812.2521] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The Viscosity Bound and Causality Violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
S. Cremonini, The Shear Viscosity to Entropy Ratio: A Status Report, Mod. Phys. Lett. B 25 (2011) 1867 [arXiv:1108.0677] [INSPIRE].
A. Rebhan and D. Steineder, Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma, Phys. Rev. Lett. 108 (2012) 021601 [arXiv:1110.6825] [INSPIRE].
K.A. Mamo, Holographic RG flow of the shear viscosity to entropy density ratio in strongly coupled anisotropic plasma, JHEP 10 (2012) 070 [arXiv:1205.1797] [INSPIRE].
S. Jain, N. Kundu, K. Sen, A. Sinha and S.P. Trivedi, A Strongly Coupled Anisotropic Fluid From Dilaton Driven Holography, JHEP 01 (2015) 005 [arXiv:1406.4874] [INSPIRE].
R. Critelli, S.I. Finazzo, M. Zaniboni and J. Noronha, Anisotropic shear viscosity of a strongly coupled non-Abelian plasma from magnetic branes, Phys. Rev. D 90 (2014) 066006 [arXiv:1406.6019] [INSPIRE].
S. Jain, R. Samanta and S.P. Trivedi, The Shear Viscosity in Anisotropic Phases, JHEP 10 (2015) 028 [arXiv:1506.01899] [INSPIRE].
M. Koschorreck, D. Pertot, E. Vogt and M. Köhl, Universal spin dynamics in two-dimensional Fermi gases, Nature Phys. 9 (2013) 405.
D.T. Son and A.O. Starinets, Viscosity, Black Holes and Quantum Field Theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240] [INSPIRE].
N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
J. Zaanen, Superconductivity: Why the temperature is high, Nature 430 (2004) 512.
A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
S.A. Hartnoll and J.E. Santos, Disordered horizons: Holography of randomly disordered fixed points, Phys. Rev. Lett. 112 (2014) 231601 [arXiv:1402.0872] [INSPIRE].
S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Emergent scale invariance of disordered horizons, JHEP 09 (2015) 160 [arXiv:1504.03324] [INSPIRE].
S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Thermal conductivity at a disordered quantum critical point, arXiv:1508.04435 [INSPIRE].
S.A. Hartnoll and J.E. Santos, Cold planar horizons are floppy, Phys. Rev. D 89 (2014) 126002 [arXiv:1403.4612] [INSPIRE].
A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
I.M. Hayeset al., Magnetoresistance near a quantum critical point, arXiv:1412.6484 [INSPIRE].
A. Donos and J.P. Gauntlett, Navier-Stokes Equations on Black Hole Horizons and DC Thermoelectric Conductivity, Phys. Rev. D 92 (2015) 121901 [arXiv:1506.01360] [INSPIRE].
A. Lucas, Conductivity of a strange metal: from holography to memory functions, JHEP 03 (2015) 071 [arXiv:1501.05656] [INSPIRE].
S.K. Chakrabarti, S. Chakrabortty and S. Jain, Proof of universality of electrical conductivity at finite chemical potential, JHEP 02 (2011) 073 [arXiv:1011.3499] [INSPIRE].
R.A. Davison, B. Goutéraux and S.A. Hartnoll, Incoherent transport in clean quantum critical metals, JHEP 10 (2015) 112 [arXiv:1507.07137] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
M. Blake, D. Tong and D. Vegh, Holographic Lattices Give the Graviton an Effective Mass, Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
M. Edalati, J.I. Jottar and R.G. Leigh, Transport Coefficients at Zero Temperature from Extremal Black Holes, JHEP 01 (2010) 018 [arXiv:0910.0645] [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
G.T. Horowitz and R.C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1998) 026005 [hep-th/9808079] [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].
S.S. Gubser and F.D. Rocha, The gravity dual to a quantum critical point with spontaneous symmetry breaking, Phys. Rev. Lett. 102 (2009) 061601 [arXiv:0807.1737] [INSPIRE].
T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, Holographic non-Fermi liquid fixed points, Phil. Trans. Roy. Soc. A 369 (2011) 1640 [arXiv:1101.0597] [INSPIRE].
A. Donos and S.A. Hartnoll, Universal linear in temperature resistivity from black hole superradiance, Phys. Rev. D 86 (2012) 124046 [arXiv:1208.4102] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
S.A. Hartnoll, Horizons, holography and condensed matter, arXiv:1106.4324 [INSPIRE].
M. Headrick, S. Kitchen and T. Wiseman, A New approach to static numerical relativity and its application to Kaluza-Klein black holes, Class. Quant. Grav. 27 (2010) 035002 [arXiv:0905.1822] [INSPIRE].
P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. 28 (2011) 215018 [arXiv:1104.4489] [INSPIRE].
O.J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, arXiv:1510.02804 [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
S.A. Hartnoll and D.M. Hofman, Locally Critical Resistivities from Umklapp Scattering, Phys. Rev. Lett. 108 (2012) 241601 [arXiv:1201.3917] [INSPIRE].
G.T. Horowitz and M.M. Roberts, Zero Temperature Limit of Holographic Superconductors, JHEP 11 (2009) 015 [arXiv:0908.3677] [INSPIRE].
J.M. Ziman, The General Variational Principle of Transport Theory, Can. J. Phys. 34 (1956) 1256.
K. Van Acoleyen, M. Mariën and F. Verstraete, Entanglement Rates and Area Laws, Phys. Rev. Lett. 111 (2013) 170501 [arXiv:1304.5931].
T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Casini, M. Huerta, R.C. Myers and A. Yale, Mutual information and the F-theorem, JHEP 10 (2015) 003 [arXiv:1506.06195] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
L. Alberte, M. Baggioli and O. Pujolàs, Viscosity bound violation in holographic solids and the viscoelastic response, arXiv:1601.03384 [INSPIRE].
P. Burikham and N. Poovuttikul, Shear viscosity in holography and effective theory of transport without translational symmetry, arXiv:1601.04624 [INSPIRE].
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Hartnoll, S.A., Ramirez, D.M. & Santos, J.E. Entropy production, viscosity bounds and bumpy black holes. J. High Energ. Phys. 2016, 170 (2016). https://doi.org/10.1007/JHEP03(2016)170
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DOI: https://doi.org/10.1007/JHEP03(2016)170