Abstract
The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time tramp. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and k-local (all-to-all interactions) and the Sachdev-Ye-Kitaev (SYK) model. Using numerical results, analytic estimates for random quantum circuits, and a heuristic analysis of Hamiltonian systems we find the following results. For geometrically local systems with a conservation law we find tramp is determined by the diffusion time across the system, order N2 for a 1D chain of N qubits. This is analogous to the behavior found for local one-body chaotic systems. For a k-local system like SYK the time is order log N but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find tramp ∼ log N, independent of connectivity.
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28 February 2019
We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.
28 February 2019
We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.
28 February 2019
We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.
28 February 2019
We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.
References
F. Haake, Quantum Signatures of Chaos, Springer-Verlag New York, Inc. (2006).
M. Serbyn and J.E. Moore, Spectral statistics across the many-body localization transition, Phys. Rev. B 93 (2016) 041424 [arXiv:1508.07293].
A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930 (2018) 45 [arXiv:1712.05073] [INSPIRE].
A. Chan, A. De Luca and J.T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, arXiv:1803.03841 [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].
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].
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].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [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, talks at KITP (2015) [http://online.kitp.ucsb.edu/online/entangled15/kitaev/] [http://online.kitp.ucsb.edu/online/entangled15/kitaev2/].
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200 [http://www.jetp.ac.ru/cgi-bin/dn/e_028_06_1200.pdf].
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].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, Nov. 10, 2014, and Stanford SITP seminars Nov. 11, 2014 and Dec. 18, 2014 [https://www.youtube.com/watch?v=OQ9qN8j7EZI].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [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, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
B. Altshuler and B. Shklovskii, Repulsion of energy levels and conductivity of small metal samples, Sov. Phys. JETP 64 (1986) 127 [http://www-thphys.physics.ox.ac.uk/talks/CMTjournalclub/sources/AltshulerShklovskii.pdf].
L. Erdős and A. Knowles, The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case, Commun. Math. Phys. 333 (2015) 1365.
A.M. García-García, 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].
T. Banks, L. Susskind and M.E. Peskin, Difficulties for the Evolution of Pure States Into Mixed States, Nucl. Phys. B 244 (1984) 125.
J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302 [quant-ph/0503210].
R. Oliveira, O.C.O. Dahlsten and M.B. Plenio, Efficient Generation of Generic Entanglement, Phys. Rev. Lett. 98 (2007) 130502 [quant-ph/0605126].
J. Emerson, Y.S. Weinstein, M. Saraceno, S. Lloyd and D.G. Cory, Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.
D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48 (2007) 052104 [quant-ph/0611002].
L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78 (2008) 062329 [arXiv:0807.0775].
A. Harrow and R. Low. Efficient quantum tensor product expanders and k-designs, Lect. Notes Comput. Sci. 5687 (2009) 548 [arXiv:0811.2597].
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501 [arXiv:0910.0913].
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
F. Brandao, A. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346 (2016) 397 [arXiv:1208.0692].
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].
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].
L. Banchi, D. Burgarth and M.J. Kastoryano, Driven Quantum Dynamics: Will It Blend?, Phys. Rev. X 7 (2017) 041015 [arXiv:1704.03041] [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].
V. Khemani, A. Vishwanath and D.A. Huse, Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws, arXiv:1710.09835 [INSPIRE].
J. Preskill, Quantum computing and the entanglement frontier, arXiv:1203.5813 [INSPIRE].
C. Neill et al., A blueprint for demonstrating quantum supremacy with superconducting qubits, Science 360 (2018) 195 [arXiv:1709.06678].
A.W. Harrow and A. Montanaro, Quantum computational supremacy, Nature 549 (2017) 203.
A. Bouland, B. Fefferman, C. Nirkhe, U. Vazirani Quantum Supremacy and the Complexity of Random Circuit Sampling, arXiv:1803.04402.
T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, arXiv:1710.09827 [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].
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
M.L. Mehta, Random matrices, vol. 142, Academic Press (2004).
T. Guhr, A. Müller-Groeling and H.A. Weidenmuller, Random matrix theories in quantum physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].
J. Flores, M. Horoi, M. Müller and T.H. Seligman, Spectral statistics of the two-body random ensemble revisited, Phys. Rev. E 63 (2001) 026204 [cond-mat/0006144] [INSPIRE].
A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930 (2018) 45 [arXiv:1712.05073] [INSPIRE].
L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17 (2014) 441 [arXiv:1407.1552].
E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, articlePhys. Rev. E 55 (1997) 4067 [cond-mat/9608116].
E. Brézin and S. Hikami, Extension of level-spacing universality, Phys. Rev. E 56 (1997) 264 [INSPIRE].
K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].
A. Gaikwad and R. Sinha, Spectral Form Factor in Non-Gaussian Random Matrix Theories, arXiv:1706.07439 [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].
D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
F. Haake, H.-J. Sommers and J. Weber, Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices, J. Phys. A 32 (1999) 6903.
P. Diaconis and S.N. Evans, Linear Functionals of Eigenvalues of Random Matrices, Trans. Am. Math. Soc. 353 (2001) 2615.
J.P. Keating, N. Linden and H.J. Wells, Random matrices and quantum spin chains, Markov Process. Related Fields 21 (2015) 537 [arXiv:1403.1114].
J.P. Keating, N. Linden and H.J. Wells, Spectra and Eigenstates of Spin Chain Hamiltonians, Commun. Math. Phys. 338 (2015) 81 [arXiv:1403.1121].
A. Pal and D.A. Huse, The many-body localization transition, arXiv:1003.2613.
D.J. Luitz, N. Laflorencie and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91 (2015) 081103 [arXiv:1411.0660].
K. Agarwal, E. Altman, E. Demler, S. Gopalakrishnan, D.A. Huse and M. Knap, Rare-region effects and dynamics near the many-body localization transition, Annalen Phys. 529 (2017) 1600326 [arXiv:1611.00770].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, in progress.
L.L. Ng, Heisenberg Model, Bethe Ansatz and Random Walks, Senior Honors Thesis, Harvard University (1996) [https://services.math.duke.edu/ng/math/papers/senior-thesis.pdf].
M. Karabach, G. Müller, H. Gould and J. Tobochnik, Introduction to the Bethe Ansatz I, Comput. Phys. 11 (1997) 36 [cond-mat/9809162].
J.R.G. Mendonca, Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite and related graphs, J. Phys. A 46 (2013) 295001 [arXiv:1207.4106].
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
J.T. Chalker, I.V. Lerner and R.A. Smith, Random Walks through the Ensemble: Linking Spectral Statistics with Wave-Function Correlations in Disordered Metals, Phys. Rev. Lett. 77 (1996) 554 [INSPIRE].
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Gharibyan, H., Hanada, M., Shenker, S.H. et al. Onset of random matrix behavior in scrambling systems. J. High Energ. Phys. 2018, 124 (2018). https://doi.org/10.1007/JHEP07(2018)124
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DOI: https://doi.org/10.1007/JHEP07(2018)124