Abstract
We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.
References
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27, KITP, U.S.A. (2015).
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].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [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].
W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, A supersymmetric SYK-like tensor model, JHEP 05 (2017) 062 [arXiv:1612.03851] [INSPIRE].
J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2d analogs of the SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
J. Yoon, Supersymmetric SYK model: bi-local collective superfield/supermatrix formulation, JHEP 10 (2017) 172 [arXiv:1706.05914] [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, Correlators in the \( \mathcal{N}=2 \) supersymmetric SYK model, JHEP 10 (2017) 202 [arXiv:1706.06078] [INSPIRE].
T. Kanazawa and T. Wettig, Complete random matrix classification of SYK models with \( \mathcal{N}=0,\;1 \) and 2 supersymmetry, JHEP 09 (2017) 050 [arXiv:1706.03044] [INSPIRE].
S. Förste and I. Golla, Nearly AdS 2 SUGRA and the super-Schwarzian, Phys. Lett. B 771 (2017) 157 [arXiv:1703.10969] [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].
T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory, JHEP 06 (2017) 111 [arXiv:1702.01738] [INSPIRE].
J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [INSPIRE].
N. Hunter-Jones, J. Liu and Y. Zhou, On thermalization in the SYK and supersymmetric SYK models, JHEP 02 (2018) 142 [arXiv:1710.03012] [INSPIRE].
O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, complexity and random matrices, JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].
T. Tao, Topics in random matrix theory, Graduate Studies in Mathematics, American Mathematical Society, U.S.A. (2012).
Vinayak and M. Žnidarič, Subsystem dynamics under random Hamiltonian evolution, J. Phys. A 45 (2012) 125204 [arXiv:1107.6035].
T.C. Bachlechner, D. Marsh, L. McAllister and T. Wrase, Supersymmetric vacua in random supergravity, JHEP 01 (2013) 136 [arXiv:1207.2763] [INSPIRE].
A. Altland and M.R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [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].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [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 , Phys. Rev. D 96 (2017) 066012 [arXiv:1701.06593] [INSPIRE].
M. Mehta, Random matrices, Pure and Applied Mathematics. Elsevier Science, The Netherlands (2004).
E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55 (1997) 4067 [cond-mat/9608116].
E. Brézin and S. Hikami, Random matrix theory with an external source, Springer Briefs in Mathematical Physics, Springer, Germany (2017).
A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor: unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350] [INSPIRE].
A.V. Andreev and B.L. Altshuler, Spectral statistics beyond random matrix theory, Phys. Rev. Lett. 75 (1995) 902 [cond-mat/9503141].
V.A. Marčenko and L.A. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR-Sb. 1 (1967) 457 [Mat. Sb. 72 (1967) 507].
F.J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Math. Phys. 3 (1962) 166.
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc. Jpn. 60 (1991) 3298.
A.J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41 (2008) 055308 [arXiv:0711.1017].
R.A. Davison et al., Thermoelectric transport in disordered metals without quasiparticles: the Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
K. Bulycheva, A note on the SYK model with complex fermions, JHEP 12 (2017) 069 [arXiv:1706.07411] [INSPIRE].
E. Dyer and G. Gur-Ari, 2D CFT partition functions at late times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement scrambling in 2D conformal field theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [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].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
X. Chen and A.W.W. Ludwig, Universal spectral correlations in the chaotic wave function and the development of quantum chaos, arXiv:1710.02686 [INSPIRE].
T. Iadecola and T.H. Hsieh, Floquet supersymmetry, arXiv:1710.05927.
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Hunter-Jones, N., Liu, J. Chaos and random matrices in supersymmetric SYK. J. High Energ. Phys. 2018, 202 (2018). https://doi.org/10.1007/JHEP05(2018)202
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DOI: https://doi.org/10.1007/JHEP05(2018)202
Keywords
- 2D Gravity
- AdS-CFT Correspondence
- Matrix Models
- Random Systems