Chaos, complexity, and random matrices

Autor: Nicholas Hunter-Jones, Junyu Liu, Jordan Cotler, Beni Yoshida
Jazyk: angličtina
Rok vydání: 2017
Předmět:
Zdroj: Journal of High Energy Physics, Vol 2017, Iss 11, Pp 1-60 (2017)
Journal of High Energy Physics
ISSN: 1029-8479
DOI: 10.1007/JHEP11(2017)048
Popis: 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 $\mathcal{O}(1)$ 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.
61 pages, 14 figures; v2: references added, typos fixed
Databáze: OpenAIRE
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