Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator
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Springer
Received: July 30, 2020 Accepted: October 9, 2020 Published: November 13, 2020
Koji Hashimoto,a Kyoung-Bum Huh,b Keun-Young Kimb and Ryota Watanabea a
Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan b School of Physics and Chemistry, Gwangju Institute of Science and Technology, 123 Cheomdan-gwagiro, Gwangju 61005, Korea
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford. Keywords: Gauge-gravity correspondence, Black Holes, Models of Quantum Gravity ArXiv ePrint: 2007.04746
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)068
JHEP11(2020)068
Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator
Contents 1 Introduction
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2 Exponential growth of OTOC in inverted harmonic oscillator
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3 Universality of the growth
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5 Summary and discussions
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A Error analysis of high-temperature fitting of Lyapunov exponents
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B Other operator orderings and the origin of the exponential growth
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1
Introduction
The exponential growth of out-of-time-order correlator (OTOC) [1] has attracted considerable attention these years, motivated by possible relations between black hole systems and quantum mechanical systems through the AdS/CFT correspondence [2]. The “chaos bound” [3] for the Lyapunov exponent λOTOC in thermal OTOCs in large N quantum theories at temperature T , λOTOC (T ) ≤ 2πT ,
(1.1)
is saturated when there exists a gravity dual in which the Lyapunov exponent is interpreted as a red shift factor near the black hole horizon probed by shock waves [4–6]. This indicator of the holographic principle indeed has lead [7–9] to a surprising quantum mechanical model, the Sachdev-Ye-Kitaev (SYK) model [10, 11], which admits a 2-dimensional dual gravity description. With the OTOC as the novel indicator of quantum chaos, quantum chaotic few-body systems have been probed to see whether the OTOC grows exponentially in time. The way to calculate microcanonical/ther
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