Abstract
Within holography the DC conductivity can be obtained by solving a system of Stokes equations for an auxiliary fluid living on the black hole horizon. We show that these equations can be derived from a novel variational principle involving a functional that depends on the fluid variables of interest as well as the time reversed quantities. This leads to simple derivation of the Onsager relations for the conductivity. We also obtain the relevant Stokes equations for bulk theories of gravity in four dimensions including a ϑF ∧ F term in the Lagrangian, where ϑ is a function of dynamical scalar fields. We discuss various realisations of the anomalous Hall conductivity that this term induces and also solve the Stokes equations for holographic lattices which break translations in one spatial dimension.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [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].
E. Banks, A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities and Stokes flows on black hole horizons, JHEP 10 (2015) 103 [arXiv:1507.00234] [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].
A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, DC Conductivity and Higher Derivative Gravity, Class. Quant. Grav. 34 (2017) 135015 [arXiv:1701.01389] [INSPIRE].
E. Banks, A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, Holographic thermal DC response in the hydrodynamic limit, Class. Quant. Grav. 34 (2017) 045001 [arXiv:1609.08912] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
A. Lucas, Hydrodynamic transport in strongly coupled disordered quantum field theories, New J. Phys. 17 (2015) 113007 [arXiv:1506.02662] [INSPIRE].
P. Morse and K. Ingard, Theoretical Acoustics, International series in pure and applied physics, Princeton University Press (1968) [https://books.google.co.uk/books?id=KIL4MV9IE5kC].
J.P. Gauntlett, J. Sonner and T. Wiseman, Quantum Criticality and Holographic Superconductors in M-theory, JHEP 02 (2010) 060 [arXiv:0912.0512] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].
M. Rozali, D. Smyth, E. Sorkin and J.B. Stang, Holographic Stripes, Phys. Rev. Lett. 110 (2013) 201603 [arXiv:1211.5600] [INSPIRE].
A. Donos, Striped phases from holography, JHEP 05 (2013) 059 [arXiv:1303.7211] [INSPIRE].
B. Withers, Black branes dual to striped phases, Class. Quant. Grav. 30 (2013) 155025 [arXiv:1304.0129] [INSPIRE].
A. Donos and J.P. Gauntlett, Minimally packed phases in holography, JHEP 03 (2016) 148 [arXiv:1512.06861] [INSPIRE].
T. Andrade and A. Krikun, Commensurate lock-in in holographic non-homogeneous lattices, JHEP 03 (2017) 168 [arXiv:1701.04625] [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].
N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald and N.P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82 (2010) 1539 [arXiv:0904.4154].
M.J. Duff, B.E.W. Nilsson and C.N. Pope, The Criterion for Vacuum Stability in Kaluza-Klein Supergravity, Phys. Lett. B 139 (1984) 154 [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].
N.R. Cooper, B.I. Halperin and I.M. Ruzin, Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field, Phys. Rev. B 55 (1997) 2344.
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].
M. Blake, A. Donos and N. Lohitsiri, Magnetothermoelectric Response from Holography, JHEP 08 (2015) 124 [arXiv:1502.03789] [INSPIRE].
S.A. Hartnoll and C.P. Herzog, Ohm’s Law at strong coupling: S duality and the cyclotron resonance, Phys. Rev. D 76 (2007) 106012 [arXiv:0706.3228] [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].
W. Fischler and S. Kundu, Hall Scrambling on Black Hole Horizons, Phys. Rev. D 92 (2015) 046008 [arXiv:1501.01316] [INSPIRE].
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
N. Seiberg and E. Witten, Gapped Boundary Phases of Topological Insulators via Weak Coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
M. Blake and A. Donos, Quantum Critical Transport and the Hall Angle, Phys. Rev. Lett. 114 (2015) 021601 [arXiv:1406.1659] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Parity-Violating Hydrodynamics in 2+1 Dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
O. Saremi and D.T. Son, Hall viscosity from gauge/gravity duality, JHEP 04 (2012) 091 [arXiv:1103.4851] [INSPIRE].
T. Delsate, V. Cardoso and P. Pani, Anti de Sitter black holes and branes in dynamical Chern-Simons gravity: perturbations, stability and the hydrodynamic modes, JHEP 06 (2011) 055 [arXiv:1103.5756] [INSPIRE].
T. Kimura and T. Nishioka, The Chiral Heat Effect, Prog. Theor. Phys. 127 (2012) 1009 [arXiv:1109.6331] [INSPIRE].
J.-W. Chen, N.-E. Lee, D. Maity and W.-Y. Wen, A Holographic Model For Hall Viscosity, Phys. Lett. B 713 (2012) 47 [arXiv:1110.0793] [INSPIRE].
P. Chesler, A. Lucas and S. Sachdev, Conformal field theories in a periodic potential: results from holography and field theory, Phys. Rev. D 89 (2014) 026005 [arXiv:1308.0329] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1704.05141
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Donos, A., Gauntlett, J.P., Griffin, T. et al. Holographic DC conductivity and Onsager relations. J. High Energ. Phys. 2017, 6 (2017). https://doi.org/10.1007/JHEP07(2017)006
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2017)006