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On the evolution of operator complexity beyond scrambling

Authors
  • Barbón, J.L.F.1
  • Rabinovici, E.2, 3
  • Shir, R.2
  • Sinha, R.1
  • 1 Instituto de Fisica Teorica IFT-UAM/CSIC, Cantoblanco, Madrid, 28049, Spain , Madrid (Spain)
  • 2 The Hebrew University, Racah Institute, Jerusalem, 9190401, Israel , Jerusalem (Israel)
  • 3 Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 91440, France , Bures-sur-Yvette (France)
Type
Published Article
Journal
Journal of High Energy Physics
Publisher
Springer-Verlag
Publication Date
Oct 29, 2019
Volume
2019
Issue
10
Identifiers
DOI: 10.1007/JHEP10(2019)264
Source
Springer Nature
Keywords
License
Yellow

Abstract

We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in [1] for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.

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