Abstract
Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.
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References
K. Hashimoto, K. Murata and R. Yoshii, Out-of-time-order correlators in quantum mechanics, JHEP 10 (2017) 138 [arXiv:1703.09435] [INSPIRE].
T. Akutagawa, K. Hashimoto, T. Sasaki and R. Watanabe, Out-of-time-order correlator in coupled harmonic oscillators, JHEP 08 (2020) 013 [arXiv:2004.04381] [INSPIRE].
A. Bhattacharyya, W. Chemissany, S.S. Haque, J. Murugan and B. Yan, The multi-faceted inverted harmonic oscillator: chaos and complexity, arXiv:2007.01232 [INSPIRE].
K. Hashimoto, K.-B. Huh, K.-Y. Kim and R. Watanabe, Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator, JHEP 11 (2020) 068 [arXiv:2007.04746] [INSPIRE].
E. Dyer and G. Gur-Ari, 2D CFT partition functions at late times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
C.M. Bender, K. Olaussen and P.S. Wang, Numerological analysis of the WKB approximation in large order, Phys. Rev. D 16 (1977) 1740 [INSPIRE].
F.T. Hioe, D. Macmillen and E.W. Montroll, Quantum theory of anharmonic oscillators: energy levels of a single and a pair of coupled oscillators with quartic coupling, Phys. Rept. 43 (1978) 305 [INSPIRE].
P. Romatschke, OTOC solver for the quartic quantum oscillator, https://github.com/paro8929/OTOC.
L. Carlitz, Some integrals containing products of Legendre polynomials, Arch. Math 12 (1961) 334.
NIST digital library of mathematical functions, http://dlmf.nist.gov/, release 1.0.25 (2019).
T.H. Koornwinder, Identities of nonterminating series by Zeilberger’s algorithm, math/9805010.
M. Laine and A. Vuorinen, Basics of thermal field theory, Springer, Germany (2016), arXiv:1701.01554 [INSPIRE].
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ArXiv ePrint: 2008.06056
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Romatschke, P. Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator. J. High Energ. Phys. 2021, 30 (2021). https://doi.org/10.1007/JHEP01(2021)030
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DOI: https://doi.org/10.1007/JHEP01(2021)030