Abstract
We compute dispersion relations of non-hydrodynamic and hydrodynamic modes in a non-relativistic strongly coupled two-dimensional quantum field theory. This is achieved by numerically computing quasinormal modes (QNMs) of a particular analytically known black brane solution to 3+1-dimensional Hǒrava Gravity. Hǒrava Gravity is distinguished from Einstein Gravity by the presence of a scalar field, termed the khronon, defining a preferred time-foliation. Surprisingly, for this black brane solution, the khronon fluctuation numerically decouples from all others, having its own set of purely imaginary eigenfrequencies, for which we provide an analytic expression. All other Hǒrava Gravity QNMs are expressed analytically in terms of QNMs of Einstein Gravity, in units involving the khronon coupling constants and various horizons. Our numerical computation reproduces the analytically known momentum diffusion mode, and extends the analytic expression for the sound modes to a wide range of khronon coupling values. In the eikonal limit (large momentum limit), the analytically known dispersion of QNM frequencies with the momentum is reproduced by our numerics. We provide a parametrization of all QNM frequencies to fourth order in the momentum. We demonstrate perturbative stability in a wide range of coupling constants and momenta.
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References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
D. Astefanesei and R.C. Myers, A New wrinkle on the enhancon, JHEP02 (2002) 043 [hep-th/0112133] [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].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
P. Hǒrava, Quantum Gravity at a Lifshitz Point, Phys. Rev.D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
P. Hǒrava and C.M. Melby-Thompson, Anisotropic Conformal Infinity, Gen. Rel. Grav.43 (2011) 1391 [arXiv:0909.3841] [INSPIRE].
C. Anderson, S.J. Carlip, J.H. Cooperman, P. Hǒrava, R.K. Kommu and P.R. Zulkowski, Quantizing Hǒrava-Lifshitz Gravity via Causal Dynamical Triangulations, Phys. Rev.D 85 (2012) 044027 [arXiv:1111.6634] [INSPIRE].
C. Germani, A. Kehagias and K. Sfetsos, Relativistic Quantum Gravity at a Lifshitz Point, JHEP09 (2009) 060 [arXiv:0906.1201] [INSPIRE].
S. Janiszewski and A. Karch, String Theory Embeddings of Nonrelativistic Field Theories and Their Holographic Hǒrava Gravity Duals, Phys. Rev. Lett.110 (2013) 081601 [arXiv:1211.0010] [INSPIRE].
S. Janiszewski and A. Karch, Non-relativistic holography from Hǒrava gravity, JHEP02 (2013) 123 [arXiv:1211.0005] [INSPIRE].
S. Janiszewski, Asymptotically hyperbolic black holes in Hǒrava gravity, JHEP01 (2015) 018 [arXiv:1401.1463] [INSPIRE].
R.A. Davison, S. Grozdanov, S. Janiszewski and M. Kaminski, Momentum and charge transport in non-relativistic holographic fluids from Hǒrava gravity, JHEP11 (2016) 170 [arXiv:1606.06747] [INSPIRE].
W. Sybesma and S. Vandoren, Lifshitz quasinormal modes and relaxation from holography, JHEP05 (2015) 021 [arXiv:1503.07457] [INSPIRE].
U. Gürsoy, A. Jansen, W. Sybesma and S. Vandoren, Holographic Equilibration of Nonrelativistic Plasmas, Phys. Rev. Lett.117 (2016) 051601 [arXiv:1602.01375] [INSPIRE].
S. Janiszewski and M. Kaminski, Quasinormal modes of magnetic and electric black branes versus far from equilibrium anisotropic fluids, Phys. Rev.D 93 (2016) 025006 [arXiv:1508.06993] [INSPIRE].
J. Bhattacharyya, Aspects of holography in Lorentz-violating gravity, Ph.D. Thesis, University of New Hampshire (2013).
T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev.D 64 (2001) 024028 [gr-qc/0007031] [INSPIRE].
G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett.2 (2009) 221 [arXiv:0811.1033] [INSPIRE].
J. Morgan, V. Cardoso, A.S. Miranda, C. Molina and V.T. Zanchin, Quasinormal modes of black holes in anti-de Sitter space: A Numerical study of the eikonal limit, Phys. Rev.D 80 (2009) 024024 [arXiv:0906.0064] [INSPIRE].
J.F. Fuini, C.F. Uhlemann and L.G. Yaffe, Damping of hard excitations in strongly coupled N = 4 plasma, JHEP12 (2016) 042 [arXiv:1610.03491] [INSPIRE].
T. Griffin, P. Hǒrava and C.M. Melby-Thompson, Lifshitz Gravity for Lifshitz Holography, Phys. Rev. Lett.110 (2013) 081602 [arXiv:1211.4872] [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].
E. Blauvelt, S. Cremonini, A. Hoover, L. Li and S. Waskie, Holographic model for the anomalous scalings of the cuprates, Phys. Rev.D 97 (2018) 061901 [arXiv:1710.01326] [INSPIRE].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav.26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys.2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].
M. Kaminski, K. Landsteiner, F. Pena-Benitez, J. Erdmenger, C. Greubel and P. Kerner, Quasinormal modes of massive charged flavor branes, JHEP03 (2010) 117 [arXiv:0911.3544] [INSPIRE].
E. Barausse, T. Jacobson and T.P. Sotiriou, Black holes in Einstein-aether and Hǒrava-Lifshitz gravity, Phys. Rev.D 83 (2011) 124043 [arXiv:1104.2889] [INSPIRE].
A.S. Miranda, J. Morgan and V.T. Zanchin, Quasinormal modes of plane-symmetric black holes according to the AdS/CFT correspondence, JHEP11 (2008) 030 [arXiv:0809.0297] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev.D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
I. Wolfram Research, Mathematica, Wolfram Research, Inc., Version 10.4.1.0 (2016).
J. Chakrabarty, Applied Plasticity, Second Edition, Springer (2010).
A. Jansen, Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes, Eur. Phys. J. Plus132 (2017) 546 [arXiv:1709.09178] [INSPIRE].
T. Jacobson and D. Mattingly, Einstein-Aether waves, Phys. Rev.D 70 (2004) 024003 [gr-qc/0402005] [INSPIRE].
J. Morgan, V. Cardoso, A.S. Miranda, C. Molina and V.T. Zanchin, Gravitational quasinormal modes of AdS black branes in d spacetime dimensions, JHEP09 (2009) 117 [arXiv:0907.5011] [INSPIRE].
C.P. Herzog, The Sound of M-theory, Phys. Rev.D 68 (2003) 024013 [hep-th/0302086] [INSPIRE].
J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Hydrodynamic Modes of Homogeneous and Isotropic Fluids, SciPost Phys.5 (2018) 014 [arXiv:1710.06885] [INSPIRE].
H. Kodama, A. Ishibashi and O. Seto, Brane world cosmology: Gauge invariant formalism for perturbation, Phys. Rev.D 62 (2000) 064022 [hep-th/0004160] [INSPIRE].
A.O. Starinets, Quasinormal modes of near extremal black branes, Phys. Rev.D 66 (2002) 124013 [hep-th/0207133] [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Parity-Violating Hydrodynamics in 2+1 Dimensions, JHEP05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
M. Kaminski and S. Moroz, Nonrelativistic parity-violating hydrodynamics in two spatial dimensions, Phys. Rev.B 89 (2014) 115418 [arXiv:1310.8305] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, Sci Post Phys.5 (2018) 011 [arXiv:1408.6855] [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
J. Hartong and N.A. Obers, Hǒrava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP07 (2015) 155 [arXiv:1504.07461] [INSPIRE].
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Garbiso, M., Kaminski, M. Dispersion relations in non-relativistic two-dimensional materials from quasinormal modes in Hǒrava Gravity. J. High Energ. Phys. 2019, 87 (2019). https://doi.org/10.1007/JHEP10(2019)087
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DOI: https://doi.org/10.1007/JHEP10(2019)087