Chaos, complexity, and random matrices

• Published in: The journal of high energy physics Vol. 2017; no. 11; pp. 1 - 60
• Main Authors: Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, Yoshida, Beni
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017; Springer Berlin; SpringerOpen
• Subjects: AdS-CFT Correspondence; Black Holes; Classical and Quantum Gravitation; Elementary Particles; MATHEMATICS AND COMPUTING; Matrix Models; Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Random Systems; Regular Article - Theoretical Physics; Relativity Theory; String Theory; AdS-CFT Correspondence; Random Systems; Matrix Models; Black Holes
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2017)048

A bstract 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. 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.