Abstract
The SYK model, a quantum mechanical model of N ≫ 1 Majorana fermions χ i , with a q-body, random interaction, is a novel realization of holography. It is known that the AdS2 dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the 1/N expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large N dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large q.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography, in KITP strings seminar and Entanglement 2015 program, 12 February, 7 April and 27 May 2015, http://online.kitp.ucsb.edu/online/entangled15/.
A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, arXiv:1611.04650 [INSPIRE].
A.M. García-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].
A.M. García-García and J.J.M. Verbaarschot, Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N, arXiv:1701.06593 [INSPIRE].
V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].
Y. Liu, M.A. Nowak and I. Zahed, Disorder in the Sachdev-Yee-Kitaev Model, 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].
J.M. Magan, Decoherence and Microscopic Diffusion at SYK, arXiv:1612.06765 [INSPIRE].
T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory, arXiv:1702.01738 [INSPIRE].
A.R. Kolovsky and D.L. Shepelyansky, Dynamical thermalization in isolated quantum dots and black holes, Europhys. Lett. 117 (2017) 10003 [arXiv:1612.06630] [INSPIRE].
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. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N , Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
D.J. Gross and V. Rosenhaus, A Generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [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].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
O. Parcollet, A. Georges, G. Kotliar and A. Sengupta, Overscreened multichannel SU(N ) Kondo model: Large-N solution and conformal field theory, Phys. Rev. B 58 (1998) 3794 [cond-mat/9711192] [INSPIRE].
O. Parcollet and A. Georges, Non-Fermi-liquid regime of a doped Mott insulator, Phys. Rev. B 59 (1999) 5341 [cond-mat/9806119].
A. Georges, O. Parcollet and S. Sachdev, Mean Field Theory of a Quantum Heisenberg Spin Glass, Phys. Rev. Lett. 85 (2000) 840 [cond-mat/9909239].
S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105 (2010) 151602 [arXiv:1006.3794] [INSPIRE].
S. Sachdev, Strange metals and the AdS/CFT correspondence, J. Stat. Mech. 1011 (2010) P11022 [arXiv:1010.0682] [INSPIRE].
B. Michel, J. Polchinski, V. Rosenhaus and S.J. Suh, Four-point function in the IOP matrix model, JHEP 05 (2016) 048 [arXiv:1602.06422] [INSPIRE].
E. Witten, An SYK-Like Model Without Disorder, arXiv:1610.09758 [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, arXiv:1612.03851 [INSPIRE].
V. Bonzom, L. Lionni and A. Tanasa, Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders, arXiv:1702.06944 [INSPIRE].
N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6 (1991) 2613 [INSPIRE].
J. Ambjørn, B. Durhuus and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A 6 (1991) 1133 [INSPIRE].
M. Gross, Tensor models and simplicial quantum gravity in > 2-D, Nucl. Phys. Proc. Suppl. 25A (1992) 144 [INSPIRE].
R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaré 12 (2011) 829 [arXiv:1011.2726] [INSPIRE].
V. Bonzom, R. Gurau, A. Riello and V. Rivasseau, Critical behavior of colored tensor models in the large-N limit, Nucl. Phys. B 853 (2011) 174 [arXiv:1105.3122] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
A. Almheiri and B. Kang, Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes, JHEP 10 (2016) 052 [arXiv:1606.04108] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
G. Mandal, P. Nayak and S.R. Wadia, Virasoro coadjoint orbits of SYK/tensor-models and emergent two-dimensional quantum gravity, arXiv:1702.04266 [INSPIRE].
V.A. Smirnov, Analytic Tools for Feynman Integrals, Springer Tracts Mod. Phys. 250 (2012) 1 [INSPIRE].
J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1 [INSPIRE].
V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
C.G. Callan Jr. and D.J. Gross, Bjorken scaling in quantum field theory, Phys. Rev. D 8 (1973) 4383 [INSPIRE].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, arXiv:1609.07832 [INSPIRE].
S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, Phys. Rev. B 95 (2017) 134302 [arXiv:1610.04619] [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].
N. Sannomiya, H. Katsura and Y. Nakayama, Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion, Phys. Rev. D 95 (2017) 065001 [arXiv:1612.02285] [INSPIRE].
D. Anninos, T. Anous and F. Denef, Disordered Quivers and Cold Horizons, JHEP 12 (2016) 071 [arXiv:1603.00453] [INSPIRE].
J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J.M.S. Wu, Two-point Functions in a Holographic Kondo Model, JHEP 03 (2017) 039 [arXiv:1612.02005] [INSPIRE].
D. Anninos and G.A. Silva, Solvable Quantum Grassmann Matrices, J. Stat. Mech. 1704 (2017) 043102 [arXiv:1612.03795] [INSPIRE].
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: the SYK models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
G. Turiaci and H. Verlinde, Towards a 2d QFT Analog of the SYK Model, arXiv:1701.00528 [INSPIRE].
Z. Bi, C.-M. Jian, Y.-Z. You, K.A. Pawlak and C. Xu, Instability of the non-Fermi liquid state of the Sachdev-Ye-Kitaev Model, Phys. Rev. B 95 (2017) 205105 [arXiv:1701.07081] [INSPIRE].
M. Berkooz, P. Narayan, M. Rozali and J. Simón, Comments on the Random Thirring Model, arXiv:1702.05105 [INSPIRE].
D.J. Gross and V. Rosenhaus, in progress.
N.S. Craigie, V.K. Dobrev and I.T. Todorov, Conformally Covariant Composite Operators in Quantum Chromodynamics, Annals Phys. 159 (1985) 411 [INSPIRE].
P.J. Forrester and S.O. Warnaar, The Importance of the Selberg Integral, Bull. Amer. Math. Soc. 45 (2008) 489 [arXiv:0710.3981].
V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A181544.
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: 1702.08016
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
Gross, D.J., Rosenhaus, V. The bulk dual of SYK: cubic couplings. J. High Energ. Phys. 2017, 92 (2017). https://doi.org/10.1007/JHEP05(2017)092
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2017)092