Chaos, complexity, and random matrices

Journal of High Energy Physics, Nov 2017

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.

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Chaos, complexity, and random matrices

HJE Chaos, complexity, and random matrices Jordan Cotler 0 1 2 4 5 Nicholas Hunter-Jones 0 1 2 5 Junyu Liu 0 1 2 5 Beni Yoshida 0 1 2 3 5 0 Pasadena , California 91125 , U.S.A 1 California Institute of Technology , USA 2 Stanford , California 94305 , U.S.A 3 Perimeter Institute for Theoretical Physics 4 Stanford Institute for Theoretical Physics, Stanford University 5 Waterloo , Ontario N2L 2Y5 , Canada Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically di cult 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 nd 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 de nition 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. AdS-CFT Correspondence; Black Holes; Matrix Models; Random Systems 1 Introduction 2 Form factors and random matrices 2.1 2.2 Random matrix theory Spectral form factors 2.2.1 2.2.2 2-point spectral form factor at in nite temperature 2-point spectral form factor at nite temperature 2.3 4-point spectral form factor at in nite temperature 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 3 Out-of-time-order correlation functions Spectral form factor from OTOCs OTOCs in random matrix theory Scrambling in random matrices 4 Frame potentials and random matrices Overview of QI machinery Frame potentials for the GUE Higher k frame potentials Frame potentials at nite temperature Time scales from GUE form factors 5 6 7 Complexity and random matrices Characterization of Haar-invariance Discussion A Scrambling and 2-designs A.1 Scrambling A.2 Unitary designs A.3 Approximate 2-designs B Information scrambling in black holes C Spectral correlators and higher frame potentials C.1 Expressions for spectral correlators C.2 Expressions for higher frame potentials C.3 Expressions for Weingarten D Additional numerics D.1 Form factors and numerics D.2 Frame potentials and numerics D.3 Minimal realizations and time averaging { i { Introduction Quantum chaos is a general feature of strongly-interacting systems and has recently provided new insight into both strongly-coupled many-body systems and the quantum nature of black holes. Even though a precise de nition of quantum chaos is not at hand, understanding how chaotic dynamics process quantum information has proven valuable. For instance, Hayden and Preskill [1] considered a simple model of random unitary evolution to show that black holes rapidly process and scramble information. The suggestion that black holes are the fastest scramblers in nature [2, 3] has led to a new probe of chaos in quantum systems, namely the 4-point out-of-time-order correlation function (OTOC). Starting with the work of Shenker and Stanford [4, 5], it was shown [6] that black holes are maximally chaotic in the sense that a bound on the early time behavior of the OTOC is saturated. Seperately, Kitaev proposed a soluble model of strongly-interacting Majorana fermions [7, 8], which reproduces many features of gravity and black holes, including the saturation of the chaos bound [9, 10]. The Sachdev-Ye-Kitaev model (SYK) has since been used as a testing ground for questions about black hole information loss and scrambling. In recent work, [11] found evidence that the late time behavior of the SYK model can be described by random matrix theory, emphasizing a dynamical perspective on more standard notions of quantum chaos. Random matrix theory (RMT) has its roots in nuclear physics [12, 13] as a statistical approach to understand the spectra of heavy atomic nuclei, famously reproducing the distribution of nearest neighbor eigenvalue spacings of nuclear resonances. Random matrix theory's early success was later followed by its adoption in a number of sub elds, including large N quantum eld theory, string theory, transport in disordered quantum systems, and quantum chaos. Indeed, random matrix eigenvalue statistics have been proposed as a de ning characteristic of quantum chaos, and it is thought that a generic classically chaotic system, when quantized, has the spectral statistics of a random matrix ens (...truncated)


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Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, Beni Yoshida. Chaos, complexity, and random matrices, Journal of High Energy Physics, 2017, pp. 48, Volume 2017, Issue 11, DOI: 10.1007/JHEP11(2017)048