Spectral form factor and semi-circle law in the time direction

[1] A.M. Garc´ıa-Garc´ıa 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].

[2] J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09

(2018) 002] [arXiv:1611.04650] [INSPIRE].

[3] A. Kitaev, A simple model of quantum holography (part 1), Talks at KITP, April 07, 2015.

[4] A. Kitaev, A simple model of quantum holography (part 2), Talks at KITP, May 27, 2015.

[5] 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].

[6] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94

(2016) 106002 [arXiv:1604.07818] [INSPIRE].

– 13 –

JHEP02(2019)161

As far as we know, there is no formula to perform this summation in a closed form. However,

it turns out that the derivative of this expression can be written in a closed form.

Let us act the derivative ∂1 − ∂2 on the last expression in (A.15) with ∂k = ∂z∂k (k =

1, 2). One can easily show that

[7] 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].

[8] T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory,

JHEP 06 (2017) 111 [arXiv:1702.01738] [INSPIRE].

[9] T. Kanazawa and T. Wettig, Complete random matrix classification of SYK models with

N = 0, 1 and 2 supersymmetry, JHEP 09 (2017) 050 [arXiv:1706.03044] [INSPIRE].

[11] N. Hunter-Jones and J. Liu, Chaos and random matrices in supersymmetric SYK, JHEP 05

(2018) 202 [arXiv:1710.08184] [INSPIRE].

[12] A.M. Garc´ıa-Garc´ıa, Y. Jia and J.J.M. Verbaarschot, Universality and Thouless energy in

the supersymmetric Sachdev-Ye-Kitaev Model, Phys. Rev. D 97 (2018) 106003

[arXiv:1801.01071] [INSPIRE].

[13] H. Gharibyan, M. Hanada, S.H. Shenker and M. Tezuka, Onset of Random Matrix Behavior

in Scrambling Systems, JHEP 07 (2018) 124 [arXiv:1803.08050] [INSPIRE].

[14] T. Nosaka, D. Rosa and J. Yoon, The Thouless time for mass-deformed SYK, JHEP 09

(2018) 041 [arXiv:1804.09934] [INSPIRE].

[15] P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity,

arXiv:1806.06840 [INSPIRE].

[16] E. Br´ezin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55

(1997) 4067 [cond-mat/9608116].

[17] J. Liu, Spectral form factors and late time quantum chaos, Phys. Rev. D 98 (2018) 086026

[arXiv:1806.05316] [INSPIRE].

[18] L. Leviandier, M. Lombardi, R. Jost and J.P. Pique, Fourier Transform: A Tool to Measure

Statistical Level Properties in Very Complex Spectra, Phys. Rev. Lett. 56 (1986) 2449.

[19] 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].

[20] J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric

Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].

[21] N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory,

J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].

[22] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,

Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].

[23] S. Kawamoto, T. Kuroki and A. Miwa, Boundary condition for D-brane from Wilson loop

and gravitational interpretation of eigenvalue in matrix model in AdS/CFT correspondence,

Phys. Rev. D 79 (2009) 126010 [arXiv:0812.4229] [INSPIRE].

[24] K. Okuyama, Connected correlator of 1/2 BPS Wilson loops in N = 4 SYM, JHEP 10

(2018) 037 [arXiv:1808.10161] [INSPIRE].

– 14 –

JHEP02(2019)161

[10] A.M. Garc´ıa-Garc´ıa, B. Loureiro, A. Romero-Berm´

udez and M. Tezuka, Chaotic-Integrable

Transition in the Sachdev-Ye-Kitaev Model, Phys. Rev. Lett. 120 (2018) 241603

[arXiv:1707.02197] [INSPIRE].

[25] B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP

01 (2014) 020 [arXiv:1311.2058] [INSPIRE].

[26] E. Br´ezin, S. Hikami and A. Zee, Oscillating density of states near zero energy for matrices

made of blocks with possible application to the random flux problem, Nucl. Phys. B 464

(1996) 411 [cond-mat/9511104] [INSPIRE].

[27] J.J.M. Verbaarschot and I. Zahed, Spectral density of the QCD Dirac operator near zero

virtuality, Phys. Rev. Lett. 70 (1993) 3852 [hep-th/9303012] [INSPIRE].

[28] M.L. Mehta, Random Matrices, Academic Press, (2004).

[30] S. Giombi, V. Pestun and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere,

JHEP 07 (2010) 088 [arXiv:0905.0665] [INSPIRE].

[31] J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity and Random Matrices,

JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].

[32] A. Gaikwad and R. Sinha, Spectral Form Factor in Non-Gaussian Random Matrix Theories,

arXiv:1706.07439 [INSPIRE].

– 15 –

JHEP02(2019)161

[29] G. Akemann and P.H. Damgaard, Wilson loops in N =4 supersymmetric Yang-Mills theory

from random matrix theory, Phys. Lett. B 513 (2001) 179 [Erratum ibid. B 524 (2002) 400]

[hep-th/0101225] [INSPIRE].

...