Abstract
We show that there exists a class of charged BTZ-like black hole solutions in Lifshitz spacetime with a hyperscaling violating factor. The charged BTZ black hole is characterized by a charge-dependent logarithmic term in the metric function. As concrete examples, we give five such charged BTZ-like black hole solutions and the standard charged BTZ metric can be regarded as a special instance of them. In order to check the recent proposed universal relations between diffusivity and the butterfly velocity, we first compute the diffusion constants of the standard charged BTZ black holes and then extend our calculation to arbitrary dimension d, exponents z and θ. Remarkably, the case d = θ and z = 2 is a very special in that the charge diffusion Dc is a constant and the energy diffusion D e might be ill-defined, but v 2 B τ diverges. We also compute the diffusion constants for the case that the DC conductivity is finite but in the absence of momentum relaxation.
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References
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [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].
A. Kitaev, A simple model of quantum holography, in KITP strings seminar and Entanglement, 12 February, 7 April and 27 May 2015, http://online.kitp.ucsb.edu/online/entangled15/.
A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, 10 November 2014, http://online.kitp.ucsb.edu/online/joint98/.
J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-Local Holography in the SYK Model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
A. Jevicki and K. Suzuki, Bi-Local Holography in the SYK Model: Perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [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].
I. Danshita, M. Hanada and M. Tezuka, Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: Towards experimental studies of quantum gravity, PTEP 2017 (2017) 083I01 [arXiv:1606.02454] [INSPIRE].
D.J. Gross and V. Rosenhaus, A Generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [INSPIRE].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].
M. Berkooz, P. Narayan, M. Rozali and J. Simón, Higher Dimensional Generalizations of the SYK Model, JHEP 01 (2017) 138 [arXiv:1610.02422] [INSPIRE].
W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].
W. Fu and S. Sachdev, Numerical study of fermion and boson models with infinite-range random interactions, Phys. Rev. B 94 (2016) 035135 [arXiv:1603.05246] [INSPIRE].
L. García- Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner and E. Solano, Digital Quantum Simulation of Minimal AdS/CFT, Phys. Rev. Lett. 119 (2017) 040501 [arXiv:1607.08560] [INSPIRE].
S.A. Hartnoll, L. Huijse and E.A. Mazenc, Matrix Quantum Mechanics from Qubits, JHEP 01 (2017) 010 [arXiv:1608.05090] [INSPIRE].
T. Nishinaka and S. Terashima, A note on Sachdev-Ye-Kitaev like model without random coupling, Nucl. Phys. B 926 (2018) 321 [arXiv:1611.10290] [INSPIRE].
G. Turiaci and H. Verlinde, Towards a 2d QFT Analog of the SYK Model, JHEP 10 (2017) 167 [arXiv:1701.00528] [INSPIRE].
S.-K. Jian and H. Yao, Solvable Sachdev-Ye-Kitaev models in higher dimensions: from diffusion to many-body localization, Phys. Rev. Lett. 119 (2017) 206602 [arXiv:1703.02051] [INSPIRE].
A. Chew, A. Essin and J. Alicea, Approximating the Sachdev-Ye-Kitaev model with Majorana wires, Phys. Rev. B 96 (2017) 121119 [arXiv:1703.06890] [INSPIRE].
E. Witten, An SYK-Like Model Without Disorder, arXiv:1610.09758 [INSPIRE].
R. Gurau, The complete 1/N expansion of a SYK-like tensor model, Nucl. Phys. B 916 (2017) 386 [arXiv:1611.04032] [INSPIRE].
I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, A Supersymmetric SYK-like Tensor Model, JHEP 05 (2017) 062 [arXiv:1612.03851] [INSPIRE].
F. Ferrari, The Large D Limit of Planar Diagrams, arXiv:1701.01171 [INSPIRE].
H. Itoyama, A. Mironov and A. Morozov, Rainbow tensor model with enhanced symmetry and extreme melonic dominance, Phys. Lett. B 771 (2017) 180 [arXiv:1703.04983] [INSPIRE].
C. Peng, Vector models and generalized SYK models, JHEP 05 (2017) 129 [arXiv:1704.04223] [INSPIRE].
Y.-Z. You, A.W.W. Ludwig and C. Xu, Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States, Phys. Rev. B 95 (2017) 115150 [arXiv:1602.06964] [INSPIRE].
A.M. Garcıa-García and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].
Y. Liu, M.A. Nowak and I. Zahed, Disorder in the Sachdev-Yee-Kitaev Model, Phys. Lett. B 773 (2017) 647 [arXiv:1612.05233] [INSPIRE].
C. Krishnan, S. Sanyal and P.N. Bala Subramanian, Quantum Chaos and Holographic Tensor Models, JHEP 03 (2017) 056 [arXiv:1612.06330] [INSPIRE].
A.M. García-García and J.J.M. Verbaarschot, Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N, Phys. Rev. D 96 (2017) 066012 [arXiv:1701.06593] [INSPIRE].
T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory, JHEP 06 (2017) 111 [arXiv:1702.01738] [INSPIRE].
R.-G. Cai, S.-M. Ruan, R.-Q. Yang and Y.-L. Zhang, The String Worldsheet as the Holographic Dual of SYK State, arXiv:1709.06297 [INSPIRE].
S.-K. Jian, Z.-Y. Xian and H. Yao, Quantum criticality and duality in the SYK/AdS 2 chain, arXiv:1709.02810 [INSPIRE].
C. Krishnan, K.V.P. Kumar and S. Sanyal, Random Matrices and Holographic Tensor Models, JHEP 06 (2017) 036 [arXiv:1703.08155] [INSPIRE].
S.R. Das, A. Jevicki and K. Suzuki, Three Dimensional View of the SYK/AdS Duality, JHEP 09 (2017) 017 [arXiv:1704.07208] [INSPIRE].
E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
J.C. Zhang et al., Anomalous Thermal Diffusivity in Underdoped YBa 2 Cu 3 O 6+x, Proc. Nat. Acad. Sci. 114 (2017) 5378 [arXiv:1610.05845] [INSPIRE].
M. Blake, Universal Charge Diffusion and the Butterfly Effect in Holographic Theories, Phys. Rev. Lett. 117 (2016) 091601 [arXiv:1603.08510] [INSPIRE].
S.-F. Wu, B. Wang, X.-H. Ge and Y. Tian, Collective diffusion and strange-metal transport, arXiv:1702.08803 [INSPIRE].
S.-F. Wu, B. Wang, X.-H. Ge and Y. Tian, Universal diffusion in holography, arXiv:1706.00718 [INSPIRE].
K.-Y. Kim and C. Niu, Diffusion and Butterfly Velocity at Finite Density, JHEP 06 (2017) 030 [arXiv:1704.00947] [INSPIRE].
X.-H. Feng and H. Lü, Butterfly Velocity Bound and Reverse Isoperimetric Inequality, Phys. Rev. D 95 (2017) 066001 [arXiv:1701.05204] [INSPIRE].
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].
L. Cheng, X.-H. Ge and S.-J. Sin, Anisotropic plasma with a chemical potential and scheme-independent instabilities, Phys. Lett. B 734 (2014) 116 [arXiv:1404.1994] [INSPIRE].
L. Cheng, X.-H. Ge and S.-J. Sin, Anisotropic plasma at finite U(1) chemical potential, JHEP 07 (2014) 083 [arXiv:1404.5027] [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].
X.-H. Ge, Y. Ling, C. Niu and S.-J. Sin, Thermoelectric conductivities, shear viscosity and stability in an anisotropic linear axion model, Phys. Rev. D 92 (2015) 106005 [arXiv:1412.8346] [INSPIRE].
Y.-L. Wang and X.-H. Ge, Shear Viscosity to Entropy Density Ratio in Higher Derivative Gravity with Momentum Dissipation, Phys. Rev. D 94 (2016) 066007 [arXiv:1605.07248] [INSPIRE].
X.-H. Ge, Notes on shear viscosity bound violation in anisotropic models, Sci. China Phys. Mech. Astron. 59 (2016) 630401 [arXiv:1510.06861] [INSPIRE].
Y. Ling, Z.-Y. Xian and Z. Zhou, Holographic Shear Viscosity in Hyperscaling Violating Theories without Translational Invariance, JHEP 11 (2016) 007 [arXiv:1605.03879] [INSPIRE].
L.Q. Fang, X.-H. Ge, J.-P. Wu and H.-Q. Leng, Anisotropic Fermi surface from holography, Phys. Rev. D 91 (2015) 126009 [arXiv:1409.6062] [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].
M. Baggioli and W.-J. Li, Diffusivities bounds and chaos in holographic Horndeski theories, JHEP 07 (2017) 055 [arXiv:1705.01766] [INSPIRE].
A. Lucas and J. Steinberg, Charge diffusion and the butterfly effect in striped holographic matter, JHEP 10 (2016) 143 [arXiv:1608.03286] [INSPIRE].
W.-J. Li, P. Liu and J.-P. Wu, Weyl corrections to diffusion and chaos in holography, arXiv:1710.07896 [INSPIRE].
A. Mokhtari, S.A. Hosseini Mansoori and K. Bitaghsir Fadafan, Diffusivities bounds in the presence of Weyl corrections, arXiv:1710.03738 [INSPIRE].
D. Ahn, Y. Ahn, H.-S. Jeong, K.-Y. Kim, W.-J. Li and C. Niu, Thermal diffusivity and butterfly velocity in anisotropic Q-Lattice models, arXiv:1708.08822 [INSPIRE].
D.-W. Pang, A Note on Black Holes in Asymptotically Lifshitz Spacetime, Commun. Theor. Phys. 62 (2014) 265 [arXiv:0905.2678] [INSPIRE].
D.-W. Pang, On Charged Lifshitz Black Holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [INSPIRE].
J. Tarrio and S. Vandoren, Black holes and black branes in Lifshitz spacetimes, JHEP 09 (2011) 017 [arXiv:1105.6335] [INSPIRE].
J. Sonner, On universality of charge transport in AdS/CFT, JHEP 07 (2013) 145 [arXiv:1304.7774] [INSPIRE].
X.-H. Ge, S.-J. Sin and S.-F. Wu, Universality of DC Electrical Conductivity from Holography, Phys. Lett. B 767 (2017) 63 [arXiv:1512.01917] [INSPIRE].
X.-H. Ge, Y. Tian, S.-Y. Wu, S.-F. Wu and S.-F. Wu, Linear and quadratic in temperature resistivity from holography, JHEP 11 (2016) 128 [arXiv:1606.07905] [INSPIRE].
S. Cremonini, H.-S. Liu, H. Lü and C.N. Pope, DC Conductivities from Non-Relativistic Scaling Geometries with Momentum Dissipation, JHEP 04 (2017) 009 [arXiv:1608.04394] [INSPIRE].
X.-H. Ge, Y. Tian, S.-Y. Wu and S.-F. Wu, Hyperscaling violating black hole solutions and Magneto-thermoelectric DC conductivities in holography, Phys. Rev. D 96 (2017) 046015 [arXiv:1606.05959] [INSPIRE].
X.-M. Kuang, E. Papantonopoulos, J.-P. Wu and Z. Zhou, The Lifshitz black branes and DC transport coefficients in massive Einstein-Maxwell-dilaton gravity, arXiv:1709.02976 [INSPIRE].
W.-J. Jiang, H.-S. Liu, H. Lü and C.N. Pope, DC Conductivities with Momentum Dissipation in Horndeski Theories, JHEP 07 (2017) 084 [arXiv:1703.00922] [INSPIRE].
S.A. Hosseini Mansoori and M.M. Qaemmaqami, Complexity Growth, Butterfly Velocity and Black hole Thermodynamics, arXiv:1711.09749 [INSPIRE].
H.-S. Liu, H. Lü and C.N. Pope, Holographic Heat Current as Noether Current, JHEP 09 (2017) 146 [arXiv:1708.02329] [INSPIRE].
M. Baggioli, Gravity, holography and applications to condensed matter, arXiv:1610.02681 [INSPIRE].
M. Alishahiha, A. Davody, A. Naseh and S.F. Taghavi, On Butterfly effect in Higher Derivative Gravities, JHEP 11 (2016) 032 [arXiv:1610.02890] [INSPIRE].
M.M. Qaemmaqami, Butterfly effect in 3D gravity, Phys. Rev. D 96 (2017) 106012 [arXiv:1707.00509] [INSPIRE].
K. Jensen, Chiral anomalies and AdS/CMT in two dimensions, JHEP 01 (2011) 109 [arXiv:1012.4831] [INSPIRE].
W. Cai, X.-H. Ge and G.-H. Yang, Diffusion in higher dimensional SYK model with complex fermions, arXiv:1711.07903 [INSPIRE].
S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].
Y. Ling, P. Liu and J.-P. Wu, Holographic Butterfly Effect at Quantum Critical Points, JHEP 10 (2017) 025 [arXiv:1610.02669] [INSPIRE].
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Ge, XH., Sin, SJ., Tian, Y. et al. Charged BTZ-like black hole solutions and the diffusivity-butterfly velocity relation. J. High Energ. Phys. 2018, 68 (2018). https://doi.org/10.1007/JHEP01(2018)068
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DOI: https://doi.org/10.1007/JHEP01(2018)068