Quantum chaos, thermalization, and entanglement generation in real-time simulations of the Banks-Fischler-Shenker-Susskind matrix model

Abstract

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 (MSS) 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 time scale 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 non-zero down to the lowest temperatures.

Chaos from massive deformations of Yang-Mills matrix models

“…In contrast, early (as well as some relatively recent) literature on the BFSS model and its deformations at finite temperature resort to the imaginary time formalism, and aim at obtaining a description of their phase diagram via both analytical perturbative and numerical (Monte Carlo) calculations [34][35][36][37][38][39]. However, very recently, adaptation of real-time techniques used in the quantum chemistry literature for solving many-body problems [46,47] proved to be applicable to both BFSS type matrix models and also on the general type of effective Hamiltonians studied in this paper with very interesting consequences [10,11]. The essence of this new approach and its relevance for our present work can be briefly described as follows.…”

“…Last but not least, another point that deserves attention is the relation between these developments and the recent as well as somewhat earlier progress in the quantum mechanical treatment of the BFSS, BMN and related matrix models via both analytical perturbative and non-perturbative Monte Carlo methods in the Euclidean time formalism [34][35][36][37][38][39]. The ties between Euclidean time formalism and the Gaussian state method in terms of their predictions for the observables are briefly discussed in sections 5 in the light of the results presented in [11] and [39]. We also suggest how future work can be oriented within this context to discover links between the temperature dependence of LLEs in real-time and the transitions between different phases of matrix models determined in the Euclidean time approach.…”

“…A natural question that arises is how to go beyond the classical descriptions given in [4] for the pure BFSS model and also its massive deformations such as those studied in the present paper. Indeed, there are quite interesting very recent advances in accessing the quantum dynamics of the BFSS model in a real-time formulation [10,11]. In the conclusions and outlook section, we provide a detailed description of the Gaussian state approximation method introduced in these references, whose main point is to introduce a general time-dependent Gaussian function, characterized by the wave packet centers and dispersions, to truncate, via the use of Wick's theorem, the otherwise infinite set of equations that arise for the expectation values of polynomial operators with increasing degree in the canonical variables, starting from the expectation values of the Heisenberg equations of motion for the latter.…”

“…Recently, there has been intense interest in exploring, modeling and understanding the emergence of chaos from matrix models of Yang-Mills(YM) theories [1][2][3][4][5][6][7][8][9][10][11][12]. Attention has been focused on examining the many aspects and features of chaos in the matrix models of primary interest within the context of M-theory and string theories, namely in the BFSS and BMN models [13][14][15][16][17][18][19].…”

Phenomenological implications of asymmetric AdS5 shock wave collision studies for heavy ion physics

“…In [30] the BFSS matrix model was investigated numerically and it was found, as in all other investigations known to us, that the leading exponent for a quantum theory stays below the MSS bound, often even substantially so.…”