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Towards the fast scrambling conjecture

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Abstract

Many proposed quantum mechanical models of black holes include highly non-local interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. In this article, we address the conjecture from two directions. First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph. Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss. Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian. The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.

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Author information

Authors and Affiliations

  1. Department of Physics, McGill University, Rue University, Montreal, QC, Canada

    Nima Lashkari & Patrick Hayden

  2. Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Via Pueblo Mall, Stanford, CA, U.S.A

    Douglas Stanford

  3. Department of Physics, Duke University, Durham, NC, U.S.A

    Matthew Hastings

  4. Microsoft Station Q, Elings Hall, Santa Barbara, CA, U.S.A

    Matthew Hastings

  5. Institut für Theoretische Physik, Appelstrasse, Hannover, Germany

    Tobias Osborne

  6. School of Computer Science, McGill University, Rue University, Montreal, QC, Canada

    Patrick Hayden

  7. Perimeter Institute for Theoretical Physics, Caroline Street, Waterloo, ON, Canada

    Patrick Hayden

Authors
  1. Nima Lashkari
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  2. Douglas Stanford
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  3. Matthew Hastings
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  4. Tobias Osborne
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  5. Patrick Hayden
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Corresponding author

Correspondence to Douglas Stanford.

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ArXiv ePrint: 1111.6580

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Lashkari, N., Stanford, D., Hastings, M. et al. Towards the fast scrambling conjecture. J. High Energ. Phys. 2013, 22 (2013). https://doi.org/10.1007/JHEP04(2013)022

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