We study numerically the onset of chaos and thermalization in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with and without fermions, considering Lyapunov exponents, entanglement generation, and quasinormal ringing.We approximate the real-time dynamics in terms of the most general Gaussian density matrices with parameters which obey self-consistent equations of motion, thus extending the applicability of real-time simulations beyond the classical limit.Initial values of these Gaussian density matrices are optimized to be as close as possible to the thermal equilibrium state of the system.Thus attempting to bridge between the low-energy regime with a calculable holographic description and the classical regime at high energies, we find that quantum corrections to classical dynamics tend to decrease the Lyapunov exponents, which is essential for consistency with the Maldacena-Shenker-Stanford bound at low temperatures.The entanglement entropy is found to exhibit an expected "scrambling" behavior-rapid initial growth followed by saturation.At least at high temperatures the entanglement saturation time appears to be governed by classical Lyapunov exponents.Decay of quasinormal modes is found to be characterized by the shortest timescale of all.We also find that while the bosonic matrix model becomes nonchaotic in the low-temperature regime, for the full BFSS model with fermions the leading Lyapunov exponent, entanglement saturation time, and decay rate of quasinormal modes all remain finite and nonzero down to the lowest temperatures.
We study the real time quantum dynamics of a matrix model consisting two bosonic fields on the fuzzy sphere using the Gaussian state approximation. Starting from the Hamiltonian formulation and using Wick's theorem, we derive a closed set of coupled nonlinear differential equations governing the time evolution of the one- and two-point correlation functions. Thermal equation of state is found by maximizing the von Neumann entropy over Gaussian states and solving algebraic self-consistency equation(s) leading to a complete determination of the symplectic spectrum of the covariance matrix. We identify near thermal initial conditions and use them to solve the equations of motion and employ our findings to probe chaos by calculating the largest Lyapunov exponent at various temperatures. Our results demonstrate that the latter tends to zero at a finite temperature indicating that the quantum dynamics respect the Maldacena,Shenker,Stanford bound across all temperatures, while approaching toward the classically chaotic regime at high temperatures. Finally, we examine the entanglement dynamics of the model in real-time by considering a sequence of bipartitions of the Hilbert space and computing the entanglement entropy and clearly exhibit the fast scrambling features that emerge in due detail.
We use exact diagonalization to study energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension ${\stackrel{^}{H}}_{S}={\stackrel{^}{H}}_{B}\ensuremath{\bigotimes}I+{\stackrel{^}{x}}_{1}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{1}+{\stackrel{^}{x}}_{2}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{3}$ of the bosonic Hamiltonian ${\stackrel{^}{H}}_{B}={\stackrel{^}{p}}_{1}^{2}+{\stackrel{^}{p}}_{2}^{2}+{\stackrel{^}{x}}_{1}^{2}{\stackrel{^}{x}}_{2}^{2}$. For a long time, this bosonic Hamiltonian was considered one of the simplest systems which exhibit dynamical chaos both classically and quantum-mechanically. Its structure closely resembles that of spatially compactified pure Yang-Mills theory. Correspondingly, the structure of our supersymmetric Hamiltonian is similar to that of spatially compactified supersymmetric Yang-Mills theory, also known as the Banks-Fischler-Shenker-Susskind (BFSS) model. We present numerical evidence that a continuous energy spectrum of the supersymmetric model leads to monotonous growth of OTOCs down to the lowest temperatures, a property that is also expected for the BFSS model from holographic duality. We find that this growth is saturated by low-energy eigenstates with effectively one-dimensional wave functions and a completely nonchaotic energy level distribution. We observe a sharp boundary separating these low-energy states from the bulk of chaotic high-energy states. Our data suggest, although with limited confidence, that at low temperatures the OTOC growth might be exponential over a finite range of time, with the corresponding Lyapunov exponent scaling linearly with temperature. In contrast, the gapped low-energy spectrum of the bosonic Hamiltonian leads to oscillating OTOCs at low temperatures without any signatures of exponential growth. We also find that the OTOCs for the bosonic Hamiltonian are never sufficiently close to the classical Lyapunov distance. On the other hand, the OTOCs for the supersymmetric system agree with the classical limit reasonably well over a finite range of temperatures and evolution times.
We analyze in detail a sharp transition between the low-energy, low-dimensional eigenstates and the high-energy chaotic bulk of the spectrum for a simple supersymmetric quantum-mechanical model with Hamiltonian
H^S=(p^12+p^22+x^12x^22)⊗I+x^1⊗σ1+x^2⊗σ3, which mimics the structure of the Banks-Fischler-Susskind-Stanford (BFSS) matrix model, the spatially compactified
N=1 super-Yang-Mills theory. We conjecture that this transition might be similar to the transition between the
D0-brane and
M-theory regimes in the BFSS model, and find that it does not lead to irregularities in the thermodynamic equation of state. We demonstrate that real-time spectral form-factor for our supersymmetric model exhibits the ``ramp'' behavior typical for quantum chaos. We also analyze the entanglement entropy and the spectrum of the reduced density matrix of the eigenstates of
H^S, considering one of the bosonic degrees of freedom as a subsystem. The entanglement entropy of low-energy eigenstates appears to be practically energy-independent. Exactly at the onset of random-matrix-type level spacing fluctuations, this behavior rapidly changes into a steady growth of entanglement with energy. We demonstrate that the spectrum of the reduced density matrix also exhibits universal level-spacing fluctuations towards its higher end, even for the ground state of the supersymmetric model. Thus even the regularly spaced, non-chaotic eigenstates contain some information about semi-classical chaotic dynamics at high energies.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate
Λ as a function of the number of degrees of freedom,
f, and inverse temperature,
β.
We show that the scaled growth rate,
Λ/f, can be described by a universal function of
fβ and displays a cross-over from classical to quantum behavior as we increase
f and/or lower the temperature. In the deep quantum regime of infinite
f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.
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.
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.