Universal spectral form factor for chaotic dynamics

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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Spectral form factor for time-dependent matrix model

Journal of High Energy Physics ◽

10.1007/jhep03(2021)071 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Arkaprava Mukherjee ◽

Shinobu Hikami

Keyword(s):

Form Factor ◽

Matrix Model ◽

Random Matrix ◽

Quantum Chaos ◽

Numerical Evaluation ◽

Time Dependent ◽

Spectral Form ◽

Random Matrix Model ◽

Large Size ◽

Analytic Expression

Abstract The quantum chaos is related to a Gaussian random matrix model, which shows a dip-ramp-plateau behavior in the spectral form factor for the large size N. The spectral form factor of time dependent Gaussian random matrix model shows also dip-ramp-plateau behavior with a rounding behavior instead of a kink near Heisenberg time. This model is converted to two matrix model, made of M1 and M2. The numerical evaluation for finite N and analytic expression in the large N are compared for the spectral form factor.

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Integrability and spectral form factor in Chern–Simons formulation

International Journal of Modern Physics A ◽

10.1142/s0217751x20501432 ◽

2020 ◽

Vol 35 (24) ◽

pp. 2050143

Author(s):

Chen-Te Ma ◽

Hongfei Shu

Keyword(s):

Form Factor ◽

Random Matrix ◽

Integrable Model ◽

Toda Chain ◽

Simons Theory ◽

Spectral Form ◽

Higher Spin ◽

Chain Theory ◽

Matrix Spectrum ◽

Chern Simons

We study the integrability from the spectral form factor in the Chern–Simons formulation. The effective action in the higher spin sector was not derived so far. Therefore, we begin from the SL(3) Chern–Simons higher spin theory. Then the dimensional reduction in this Chern–Simons theory gives the SL(3) reparametrization invariant Schwarzian theory, which is the boundary theory of an interacting theory between the spin-2 and spin-3 fields at the infrared or massless limit. We show that the Lorentzian SL(3) Schwarzian theory is dual to the integrable model, SL(3) open Toda chain theory. Finally, we demonstrate the application of open Toda chain theory from the SL(2) case. The numerical result shows that the spectral form factor loses the dip-ramp-plateau behavior, consistent with integrability. The spectrum is not a Gaussian random matrix spectrum. We also give an exact solution of the spectral form factor for the SL(3) theory. This solution provides a similar form to the SL(2) case for [Formula: see text]. Hence the SL(3) theory should also do not have a Gaussian random matrix spectrum.

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