Abstract
We describe a numerical method which allows to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.
Highlights
Thermalization of strongly interacting quantum systems is one of the important problems in different areas of modern physics, ranging from apparent thermalization of the quark-gluon plasma [1,2,3] in heavy-ion collisions to inflation of our Universe and dynamics of ultra-cold quantum gases [4]
In the regime of large field strengths and small coupling constants one can reliably use the classical equations of motion [5, 6], re-summing secular divergences by averaging over quantum fluctuations in initial conditions [5]
In these Proceedings we have outlined the Gaussian state approximation for the real-time dynamics of many-body systems, used previously in the quantum chemistry context [8], and demonstrated on a simple example that it is capable of reproducing the essential features of quantum dynamics which are absent for the classical equations of motion
Summary
Thermalization of strongly interacting quantum systems is one of the important problems in different areas of modern physics, ranging from apparent thermalization of the quark-gluon plasma [1,2,3] in heavy-ion collisions to inflation of our Universe and dynamics of ultra-cold quantum gases [4]. In these Proceedings, we aim to go deeper into this intermediate regime of neither large nor small occupation numbers. The dynamics described by the Hamiltonian (1) is known to be classically chaotic, so that the distance between initially very close points in phase space grows exponentially with time [9] The fact that such chaotic classical systems effectively forget about initial conditions after some “thermalization” time can be interpreted as the formation of a black hole state [3, 10,11,12] in the framework of holographic duality between compactified super-Yang-Mills theory and the gravitationally interacting system of D0 branes [13]. In this work we use the Gaussian state approximation of [8] to understand how quantum effects might affect the classically chaotic dynamics of the Hamiltonian (1)
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