Quantum Lyapunov spectrum
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Springer
Received: December 15, 2018 Accepted: March 29, 2019 Published: April 10, 2019
Quantum Lyapunov spectrum
a
Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, U.S.A. b Department of Physics, University of Colorado, Boulder, Colorado 80309, U.S.A. c Condensed Matter Theory Center, Maryland Center for Fundamental Physics, Joint Center for Quantum Information and Computer Science, and Department of Physics, University of Maryland, College Park MD 20742, U.S.A. d Department of Physics, Kyoto University, Kyoto 606-8502, Japan
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems. Keywords: AdS-CFT Correspondence, Field Theories in Lower Dimensions, Random Systems ArXiv ePrint: 1809.01671
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)082
JHEP04(2019)082
Hrant Gharibyan,a Masanori Hanada,b Brian Swinglec and Masaki Tezukad
Contents 1
2 Lyapunov spectrum in classical chaos 2.1 Kolmogorov-Sinai entropy 2.1.1 Application to black hole 2.2 Universality in the Lyapunov spectrum 2.3 Technical remarks
3 4 4 6 6
3 Models 3.1 Sachdev-Ye-Kitaev (SYK) 3.2 XXZ spin chain
7 7 8
4 A definition: quantum Lyapunov exponents 4.1 Lyapunov exponents in SYK 4.2 Lyapunov exponents in XXZ 4.3 Kolmogorov-Sinai entropy 4.4 Is the ‘perturbation’ actually small? 4.5 Relations to other approaches
8 8 9 9 10 11
5 Lyapunov growth 5.1 Lyapunov growth in SYK 5.1.1 The largest exponent vs λ(OTOC) 5.1.2 Kolmogorov-Sinai and entanglement entropy 5.1.3 Fastest entropy generator? 5.2 Lyapunov growth in XXZ
13 13 13 17 18 19
6 Random matrix statistics of Lyapunov spectrum 6.1 Lyapunov spectrum vs RMT in SYK 6.2 Lyapunov spectrum vs RMT in XXZ
22 22 24
7 Conclusion and outlook
26
1
Introduction
Many-body quantum chaos is of fundamental interest in a variety of fields of physics, including condensed matter, quantum information, and quantum gravity. Considerable recent progress has come from the realization that it is possible, in some cases, to define a ki
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