Abstract
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.
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References
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
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, http://online.kitp.ucsb.edu/online/entangled15/, U.S.A., 12 February, 7 April and 27 May 2015.
O. Parcollet and A. Georges, Non-Fermi-liquid regime of a doped Mott insulator, Phys. Rev. B 59 (1999) 5341 [cond-mat/9806119].
Y. Liu, M.A. Nowak and I. Zahed, Disorder in the Sachdev-Yee-Kitaev model, arXiv:1612.05233 [INSPIRE].
P. Betzios, U. Gürsoy and O. Papadoulaki, Matrix quantum mechanics on \( {S}^1/{\mathrm{\mathbb{Z}}}_2 \), arXiv:1612.04792 [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, A supersymmetric SYK-like tensor model, arXiv:1612.03851 [INSPIRE].
D. Anninos and G.A. Silva, Solvable quantum Grassmann matrices, arXiv:1612.03795 [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, arXiv:1612.02005 [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, arXiv:1612.00849 [INSPIRE].
T. Nishinaka and S. Terashima, A note on Sachdev-Ye-Kitaev like model without random coupling, arXiv:1611.10290 [INSPIRE].
M. Blake and A. Donos, Diffusion and chaos from near AdS 2 horizons, JHEP 02 (2017) 013 [arXiv:1611.09380] [INSPIRE].
J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J.M.S. Wu, Holographic Kondo and Fano resonances, arXiv:1611.09368 [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, arXiv:1611.04650 [INSPIRE].
X. Chen, T. Zhou, D.A. Huse and E. Fradkin, Out-of-time-order correlations in many-body localized and thermal phases, arXiv:1610.00220 [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].
Y. Ling, P. Liu and J.-P. Wu, Note on the butterfly effect in holographic superconductor models, arXiv:1610.07146 [INSPIRE].
K. Hashimoto and N. Tanahashi, Universality in chaos of particle motion near black hole horizon, Phys. Rev. D 95 (2017) 024007 [arXiv:1610.06070] [INSPIRE].
S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, 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].
Y. Ling, P. Liu and J.-P. Wu, Holographic butterfly effect at quantum critical points, arXiv:1610.02669 [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, arXiv:1609.07832 [INSPIRE].
N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95 (2017) 012120 [arXiv:1609.00015] [INSPIRE].
S. Giombi, I.R. Klebanov and Z.M. Tan, The ABC of higher-spin AdS/CFT, arXiv:1608.07611 [INSPIRE].
A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [INSPIRE].
D. Radicevic, Quantum mechanics in the infrared, arXiv:1608.07275 [INSPIRE].
S.A. Hartnoll, L. Huijse and E.A. Mazenc, Matrix quantum mechanics from qubits, JHEP 01 (2017) 010 [arXiv:1608.05090] [INSPIRE].
L. García-Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner and E. Solano, Digital quantum simulation of minimal AdS/CFT, arXiv:1607.08560 [INSPIRE].
N.Y. Yao et al., Interferometric approach to probing fast scrambling, arXiv:1607.01801 [INSPIRE].
G. Zhu, M. Hafezi and T. Grover, Measurement of many-body chaos using a quantum clock, Phys. Rev. A 94 (2016) 062329 [arXiv:1607.00079] [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].
K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
D.A. Roberts and B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
G. Turiaci and H. Verlinde, On CFT and quantum chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE].
D. Anninos, T. Anous and F. Denef, Disordered quivers and cold horizons, JHEP 12 (2016) 071 [arXiv:1603.00453] [INSPIRE].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
Y. Gu and X.-L. Qi, Fractional statistics and the butterfly effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, arXiv:1611.04650 [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].
R. Gurau, Invitation to random tensors, SIGMA 12 (2016) 094 [arXiv:1609.06439] [INSPIRE].
V. Bonzom, R. Gurau and V. Rivasseau, Random tensor models in the large-N limit: uncoloring the colored tensor models, Phys. Rev. D 85 (2012) 084037 [arXiv:1202.3637] [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].
R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré 13 (2012) 399 [arXiv:1102.5759] [INSPIRE].
R. Gurau and V. Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, Europhys. Lett. 95 (2011) 50004 [arXiv:1101.4182] [INSPIRE].
R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaré 12 (2011) 829 [arXiv:1011.2726] [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].
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].
F. Haake, Quantum signatures of chaos, Springer, Germany, (2010).
P. Shukla, Random matrix theory and applications, IIT Kgharagpur lectures, http://nptel.ac.in/courses/115105052/1.
C. Krishnan, Quantum field theory, black holes and holography, arXiv:1011.5875 [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, arXiv:1602.06964 [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].
A. Altland and M.R. Zirnbauer, Novel symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [cond-mat/9602137].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K., (2012).
V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, arXiv:1612.04334 [INSPIRE].
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Krishnan, C., Sanyal, S. & Subramanian, P.N.B. Quantum chaos and holographic tensor models. J. High Energ. Phys. 2017, 56 (2017). https://doi.org/10.1007/JHEP03(2017)056
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DOI: https://doi.org/10.1007/JHEP03(2017)056