Abstract
We consider the Brownian SYK model of N interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-2 tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of N qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with N , allowing us to perform numerically exact calculations up to N = 106. We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time t∗ ∼ ln N controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.
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References
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854.
L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239.
Y.D. Lensky and X.-L. Qi, Chaos and high temperature pure state thermalization, JHEP 06 (2019) 025 [arXiv:1805.03675] [INSPIRE].
P. Hosur and X.-L. Qi, Characterizing eigenstate thermalization via measures in the Fock space of operators, Phys. Rev. E 93 (2016) 042138.
T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
H. Liu and S.J. Suh, Entanglement tsunami: universal scaling in holographic thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
K.A. Landsman et al., Verified quantum information scrambling, Nature 567 (2019) 61 [arXiv:1806.02807] [INSPIRE].
G. Bentsen, Y. Gu and A. Lucas, Fast scrambling on sparse graphs, Proc. Nat. Acad. Sci. 116 (2019) 6689 [arXiv:1805.08215] [INSPIRE].
M.C. Gutzwiller, Chaos in classical and quantum mechanics, Springer, New York, NY, U.S.A. (1990).
H.-J. Stöckmann, Quantum chaos: an introduction, Cambridge University Press, Cambridge, U.K. (2007).
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk at Fundamental Physics Prize Symposium, University of California, Santa Barbara, CA, U.S.A., 12 February 2014.
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP strings seminar and Entanglement 2015 program, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP strings seminar and Entanglement 2015 program, University of California, Santa Barbara, CA, U.S.A., 27 May 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, 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].
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [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].
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: the Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [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].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].
X.-Y. Song, C.-M. Jian and L. Balents, Strongly correlated metal built from Sachdev-Ye-Kitaev models, Phys. Rev. Lett. 119 (2017) 216601 [arXiv:1705.00117] [INSPIRE].
D. Chowdhury, Y. Werman, E. Berg and T. Senthil, Translationally invariant non-Fermi liquid metals with critical Fermi-surfaces: solvable models, Phys. Rev. X 8 (2018) 031024 [arXiv:1801.06178] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
A. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200 [Zh. Eksp. Teor. Fiz. 55 (1969) 2262].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [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].
I.L. Aleiner, L. Faoro and L.B. Ioffe, Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves, Annals Phys. 375 (2016) 378 [arXiv:1609.01251] [INSPIRE].
B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].
N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95 (2017) 012120 [arXiv:1609.00015] [INSPIRE].
A.A. Patel and S. Sachdev, Quantum chaos on a critical Fermi surface, Proc. Nat. Acad. Sci. 114 (2017) 1844 [arXiv:1611.00003] [INSPIRE].
I. Kukuljan, S. Grozdanov and T. Prosen, Weak quantum chaos, Phys. Rev. B 96 (2017) 060301 [arXiv:1701.09147] [INSPIRE].
B. Dóra and R. Moessner, Out-of-time-ordered density correlators in Luttinger liquids, Phys. Rev. Lett. 119 (2017) 026802 [arXiv:1612.00614] [INSPIRE].
N. Tsuji, P. Werner and M. Ueda, Exact out-of-time-ordered correlation functions for an interacting lattice fermion model, Phys. Rev. A 95 (2017) 011601 [arXiv:1610.01251] [INSPIRE].
C.-J. Lin and O.I. Motrunich, Out-of-time-ordered correlators in a quantum Ising chain, Phys. Rev. B 97 (2018) 144304 [arXiv:1801.01636] [INSPIRE].
C.-J. Lin and O.I. Motrunich, Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the Luttinger-liquid model, Phys. Rev. B 98 (2018) 134305 [arXiv:1807.08826] [INSPIRE].
A. Smith, J. Knolle, R. Moessner and D.L. Kovrizhin, Logarithmic spreading of out-of-time-ordered correlators without many-body localization, Phys. Rev. Lett. 123 (2019) 086602 [arXiv:1812.07981] [INSPIRE].
S. Nakamura, E. Iyoda, T. Deguchi and T. Sagawa, Universal scrambling in gapless quantum spin chains, Phys. Rev. B 99 (2019) 224305 [arXiv:1904.09778] [INSPIRE].
M. McGinley, A. Nunnenkamp and J. Knolle, Slow growth of out-of-time-order correlators and entanglement entropy in integrable disordered systems, Phys. Rev. Lett. 122 (2019) 020603 [arXiv:1807.06039] [INSPIRE].
Y. Huang, F.G. S.L. Brandão and Y.-L. Zhang, Finite-size scaling of out-of-time-ordered correlators at late times, Phys. Rev. Lett. 123 (2019) 010601 [arXiv:1705.07597] [INSPIRE].
J. Chávez-Carlos et al., Quantum and classical Lyapunov exponents in atom-field interaction systems, Phys. Rev. Lett. 122 (2019) 024101 [arXiv:1807.10292] [INSPIRE].
A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7 (2017) 031016 [arXiv:1608.06950] [INSPIRE].
C. Sünderhauf, D. Pérez-García, D.A. Huse, N. Schuch and J.I. Cirac, Localization with random time-periodic quantum circuits, Phys. Rev. B 98 (2018) 134204 [arXiv:1805.08487] [INSPIRE].
A. Nahum, S. Vijay and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].
C. von Keyserlingk, T. Rakovszky, F. Pollmann and S. Sondhi, Operator hydrodynamics, OTOCs and entanglement growth in systems without conservation laws, Phys. Rev. X 8 (2018) 021013 [arXiv:1705.08910] [INSPIRE].
A. Chan, A. De Luca and J.T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X 8 (2018) 041019 [arXiv:1712.06836] [INSPIRE].
A. Chan, A. De Luca and J.T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, Phys. Rev. Lett. 121 (2018) 060601 [arXiv:1803.03841] [INSPIRE].
T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8 (2018) 031058 [arXiv:1710.09827] [INSPIRE].
V. Khemani, A. Vishwanath and D.A. Huse, Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws, Phys. Rev. X 8 (2018) 031057 [arXiv:1710.09835] [INSPIRE].
P. Kos, M. Ljubotina and T. Prosen, Many-body quantum chaos: analytic connection to random matrix theory, Phys. Rev. X 8 (2018) 021062 [arXiv:1712.02665] [INSPIRE].
B. Bertini, P. Kos and T. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett. 121 (2018) 264101 [arXiv:1805.00931] [INSPIRE].
N. Hunter-Jones, Unitary designs from statistical mechanics in random quantum circuits, arXiv:1905.12053 [INSPIRE].
M.J. Gullans and D.A. Huse, Entanglement structure of current-driven diffusive fermion systems, Phys. Rev. X 9 (2019) 021007 [arXiv:1804.00010] [INSPIRE].
T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99 (2019) 174205 [arXiv:1804.09737] [INSPIRE].
A.J. Friedman, A. Chan, A. De Luca and J.T. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, arXiv:1906.07736 [INSPIRE].
J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302.
J. Emerson, Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.
O.C.O. Dahlsten, R. Oliveira and M.B. Plenio, The emergence of typical entanglement in two-party random processes, J. Phys. A 40 (2007) 8081.
D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys. 48 (2007) 052104.
R. Oliveira, O.C.O. Dahlsten and M.B. Plenio, Generic entanglement can be generated efficiently, Phys. Rev. Lett. 98 (2007) 130502.
M. Žnidarič, Optimal two-qubit gate for generation of random bipartite entanglement, Phys. Rev. A 76 (2007) 012318.
M. Žnidarič, Exact convergence times for generation of random bipartite entanglement, Phys. Rev. A 78 (2008) 032324.
L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78 (2008) 062329.
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys. 291 (2009) 257.
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501.
I.T. Diniz and D. Jonathan, Comment on “random quantum circuits are approximate 2-designs”, Commun. Math. Phys. 304 (2011) 281.
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
F.G. S.L. Brandão, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346 (2016) 397 [arXiv:1208.0692] [INSPIRE].
Y. Nakata, C. Hirche, M. Koashi and A. Winter, Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynamics, Phys. Rev. X 7 (2017) 021006 [arXiv:1609.07021] [INSPIRE].
S. Choi, Y. Bao, X.-L. Qi and E. Altman, Quantum error correction in scrambling dynamics and measurement induced phase transition, arXiv:1903.05124 [INSPIRE].
X.-L. Qi and A. Streicher, Quantum epidemiology: operator growth, thermal effects and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [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].
A.M. García-García, Y. Jia and J.J.M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N 2 , JHEP 04 (2018) 146 [arXiv:1801.02696] [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].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
G. Gur-Ari, R. Mahajan and A. Vaezi, Does the SYK model have a spin glass phase?, JHEP 11 (2018) 070 [arXiv:1806.10145] [INSPIRE].
O. Schnaack, N. Bölter, S. Paeckel, S.R. Manmana, S. Kehrein and M. Schmitt, Tripartite information, scrambling and the role of Hilbert space partitioning in quantum lattice models, arXiv:1808.05646 [INSPIRE].
E. Iyoda and T. Sagawa, Scrambling of quantum information in quantum many-body systems, Phys. Rev. A 97 (2018) 042330 [arXiv:1704.04850] [INSPIRE].
S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva and R. Fazio, Scrambling and entanglement spreading in long-range spin chains, Phys. Rev. B 98 (2018) 134303 [arXiv:1806.00022] [INSPIRE].
A. Seshadri, V. Madhok and A. Lakshminarayan, Tripartite mutual information, entanglement and scrambling in permutation symmetric systems with an application to quantum chaos, Phys. Rev. E 98 (2018) 052205 [arXiv:1806.00113] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
F. Haake, Quantum signatures of chaos, Springer, Berlin, Heidelberg, Germany (2010).
P. Ribeiro, J. Vidal and R. Mosseri, Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections, Phys. Rev. E 78 (2008) 021106.
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
T. Zhou and X. Chen, Operator dynamics in a Brownian quantum circuit, Phys. Rev. E 99 (2019) 052212 [arXiv:1805.09307] [INSPIRE].
S. Xu and B. Swingle, Locality, quantum fluctuations and scrambling, Phys. Rev. X 9 (2019) 031048 [arXiv:1805.05376] [INSPIRE].
X. Chen and T. Zhou, Quantum chaos dynamics in long-range power law interaction systems, Phys. Rev. B 100 (2019) 064305 [arXiv:1808.09812] [INSPIRE].
H. Gharibyan, M. Hanada, S.H. Shenker and M. Tezuka, Onset of random matrix behavior in scrambling systems, JHEP 07 (2018) 124 [Erratum ibid. 02 (2019) 197] [arXiv:1803.08050] [INSPIRE].
K. Parthasarathy, An introduction to quantum stochastic calculus, Monogr. Math. 85, Birkhäuser, Basel, Switzerland (1992)
L. Banchi, D. Burgarth and M.J. Kastoryano, Driven quantum dynamics: will it blend?, Phys. Rev. X 7 (2017) 041015 [arXiv:1704.03041] [INSPIRE].
E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A.H. Werner and J. Eisert, Mixing properties of stochastic quantum Hamiltonians, Commun. Math. Phys. 355 (2017) 905 [arXiv:1606.01914] [INSPIRE].
J.R. González Alonso, N. Yunger Halpern and J. Dressel, Out-of-time-ordered-correlator quasiprobabilities robustly witness scrambling, Phys. Rev. Lett. 122 (2019) 040404 [arXiv:1806.09637] [INSPIRE].
F. Iglói and I. Peschel, On reduced density matrices for disjoint subsystems, Europhys. Lett. 89 (2010) 40001.
M. Fagotti and P. Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Statist. Mech. 2010 (2010) P04016.
Y. Gu, A. Lucas and X.-L. Qi, Spread of entanglement in a Sachdev-Ye-Kitaev chain, JHEP 09 (2017) 120 [arXiv:1708.00871] [INSPIRE].
T. Prosen and I. Pižorn, Operator space entanglement entropy in a transverse Ising chain, Phys. Rev. A 76 (2007) 032316.
J. Dubail, Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d, J. Phys. A 50 (2017) 234001 [arXiv:1612.08630] [INSPIRE].
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Sünderhauf, C., Piroli, L., Qi, XL. et al. Quantum chaos in the Brownian SYK model with large finite N : OTOCs and tripartite information. J. High Energ. Phys. 2019, 38 (2019). https://doi.org/10.1007/JHEP11(2019)038
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DOI: https://doi.org/10.1007/JHEP11(2019)038