A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble
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Springer
Received: December Revised: April Accepted: May Published: May
15, 30, 11, 23,
2018 2019 2019 2019
Sayantan Choudhurya,1,2 and Arkaprava Mukherjeeb a
Quantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨ uhlenberg 1, Potsdam-Golm 14476, Germany b Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, West Bengal 741246, India
E-mail: [email protected], [email protected] Abstract: In this article, using the principles of Random Matrix Theory (RMT) with Gaussian Unitary Ensemble (GUE), we give a measure of quantum chaos by quantifying Spectral From Factor (SFF) appearing from the computation of two point Out of Time Order Correlation function (OTOC) expressed in terms of square of the commutator bracket of quantum operators which are separated in time scale. We also provide a strict model independent bound on the measure of quantum chaos, −1/N (1 − 1/π) ≤ SFF ≤ 0 and 0 ≤ SFF ≤ 1/πN , valid for thermal systems with large and small number of degrees of freedom respectively. We have studied both the early and late behaviour of SFF to check the validity and applicability of our derived bound. Based on the appropriate physical arguments we give a precise mathematical derivation to establish this alternative strict bound of quantum chaos. Finally, we provide an example of integrability from GUE based RMT from Toda Lattice model to explicitly show the application of our derived bound on SFF to quantify chaos. Keywords: Matrix Models, Random Systems, Thermal Field Theory ArXiv ePrint: 1811.01079 1
Corresponding author. This project is the part of the non-profit virtual international research consortium “Quantum Structures of the Space-Time & Matter”. 2
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)149
JHEP05(2019)149
A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble
Contents 1
2 Random Matrix Theory (RMT) redux: quantifying quantum chaos
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3 Out of time ordered correlation function (OTOC) from RMT
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4 Two point Spectral Form Factor (SFF) from thermal Green’s function of RMT
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5 Bound on two point SFF from GUE based RMT
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6 Early time behaviour of two point SFF from GUE based RMT
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7 Integrablity from GUE based RMT: Toda Lattice model
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8 Conclusion
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1
Introduction
Quantum description of chaos has three important properties boundedness, exponential sensitivity and infinite recurrence. In the context of the study of dynamical systems the concept of quantum chaos describes quantum signatures of classically chaotic systems. Quantum chaos [1] can be formulated for two observable represented by hermitian operators X(t) and Y (t) using their commutator relation. This actually explains the perturbation effect of one operator Y (t) on the measurement of other operator X(t). Strength of this perturbation can be measured by formulating a time dependent func
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