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Title: Chaos, complexity, and random matrices

Abstract

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. Here,we envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

Authors:
 [1]; ORCiD logo [2];  [2];  [3]
  1. Stanford Univ., Stanford, CA (United States). Stanford Inst. for Theoretical Physics
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States)
  3. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
Publication Date:
Research Org.:
California Inst. of Technology, Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1501469
Grant/Contract Number:  
SC0011632
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Black Holes; Matrix Models; Random Systems

Citation Formats

Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, and Yoshida, Beni. Chaos, complexity, and random matrices. United States: N. p., 2017. Web. doi:10.1007/jhep11(2017)048.
Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, & Yoshida, Beni. Chaos, complexity, and random matrices. United States. doi:10.1007/jhep11(2017)048.
Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, and Yoshida, Beni. Thu . "Chaos, complexity, and random matrices". United States. doi:10.1007/jhep11(2017)048. https://www.osti.gov/servlets/purl/1501469.
@article{osti_1501469,
title = {Chaos, complexity, and random matrices},
author = {Cotler, Jordan and Hunter-Jones, Nicholas and Liu, Junyu and Yoshida, Beni},
abstractNote = {Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. Here,we envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.},
doi = {10.1007/jhep11(2017)048},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2017,
place = {United States},
year = {2017},
month = {11}
}

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