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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.

Our understanding of quantum chaos has significantly advanced in recent years due to numerous correspondences between chaotic systems and black holes. In particular, it was argued that physical systems which are holographically dual to black holes are maximally chaotic, with the Sachdev-Ye-Kitaev (SYK) model

Quantum chaos can be described quantitatively in terms of the exponential growth of the out-of-time-order correlators (OTOCs)

Exponential growth of OTOCs has to be contrasted with the time dependence of the conventional time-ordered correlators

While in classical systems Lyapunov exponents can be arbitrarily large, a universal Maldacena-Stanford-Shenker (MSS) bound

We have set

The BFSS model

Quantum entanglement between different degrees of freedom (d.o.f.) offers a complementary language for a quantitative description of quantum chaos. It is expected that for strongly interacting chaotic systems all d.o.f. become highly entangled under quantum evolution

In this paper we report on numerical studies of quantum corrections to the real-time dynamics of the thermal states of the BFSS model and its bosonic sector (bosonic matrix model), addressing in particular quantum corrections to Lyapunov exponents, the relation between Lyapunov exponents and entanglement entropy generation, and quasinormal ringing. We find that quantum corrections from the bosonic sector of the model tend to make the system less chaotic and less dissipative, whereas the contribution of Majorana fermions works in the opposite direction. The characteristic Lyapunov time, entanglement saturation time, and decay time of quasinormal ringing become very long for the bosonic matrix model at sufficiently low temperatures, which roughly correspond to the confinement regime

We further demonstrate that the characteristic saturation time for the entanglement entropy is in general shorter than the Lyapunov time

In order to simulate the real-time dynamics of the BFSS model, we approximate the density matrix of the system by the most general Gaussian function with time-dependent parameters which obey self-consistent equations of motion. Such an approach, which we will refer to as the Gaussian state approximation, is closely related to the semiclassical approximation

As discussed in

In the context of the BFSS model, one of the limitations of the Gaussian state approximation is that the gauge symmetry constraints cannot be fully respected. As a consequence, our simulations correspond to the ungauged version of the BFSS or bosonic matrix models, where no gauge constraints are imposed on the state vectors. Fortunately, the differences between the gauged and ungauged models appear to be minor at least at low temperatures, as conjectured recently in

We start our discussion in Sec.

In this paper we use the following representation of the Hamiltonian of the BFSS matrix model

The indices

The indices

Getting rid of explicit Lie algebra indices and treating

The representation

On the space of physical states defined by

In this section, we explain how the Gaussian state approximation is obtained by truncating the full equations of motion (Heisenberg equations)

Averaging these equations of motion over some density matrix, we can express the time derivatives of the expectation values

In order to obtain a treatable approximation to the full Heisenberg equations

Averaging Eqs.

To make our approximation self-consistent, we also need to describe the time evolution of the two-point correlators which enter Eqs.

We can again average these equations over our Gaussian density matrix and apply Wick’s theorem. This is straightforward for all equations except

Since we have assumed that the only nonzero correlator with fermions is

Symmetrizing the operator products in the expectation values

Note that equations of motion for the bosonic two-point correlators do not contain fermionic correlators. Fermions only affect the dynamics due to the coupling to the expectation values

Equations

The conservation of the generators of the gauge transformations

Fortunately, as discussed recently in

Another important property of Eqs.

Similar to the full Schrödinger equation

In contrast to the full Schrödinger equation for the Hamiltonian

In this work we consider three different approximations to the full real-time dynamics of the BFSS model

Real-time dynamics of the ungauged

Real-time dynamics of the full ungauged

Equations

Within the Gaussian state approximation the thermal density matrix by definition should also be Gaussian (i.e., correspond to a Gaussian Wigner function). If the system is in contact with a thermostat which does not perform work (e.g., collisions with a hard wall), upon thermalization the von Neumann entropy of a state should reach its maximal possible value for a given energy. Based on this very general physical principle, we will approximate the thermal equilibrium states by those Gaussian density matrices which have the largest possible von Neumann entropy at a given energy.

To this end we need to know the von Neumann entropy of an arbitrary Gaussian density matrix, which can be expressed in terms of the correlators

Symplectic eigenvalues of the matrix

The von Neumann entropy

Thermal equilibrium states should be invariant under spatial and internal

The von Neumann entropy

Substituting the correlators

In order to maximize the entropy

Now the only missing ingredient in our analysis of the equation of state is the temperature, which can be introduced using the standard thermodynamic relation

In Fig.

Equation of state for the ungauged bosonic matrix model and the ungauged BFSS model within the Gaussian state approximation compared with the results of Monte Carlo simulations

We indeed observe a rather good agreement within a few percent accuracy for the temperature dependence of both the energy and the coordinate dispersion for all simulation parameters used in

There are two possible ways to interpret the Gaussian state characterized by the correlators

A trivial way is to directly substitute the correlators

In what follows we use another, physically better motivated option of interpreting mixed Gaussian states with

We then represent the finite-temperature Gaussian density matrix characterized by correlators

Time dependence of the coordinate dispersion

This interpretation of the classical component of the dispersions of

Of course, the two interpretations discussed above would be equivalent for unitary evolution, but yield drastically different results for the nonunitary evolution within the Gaussian state approximation.

To obtain the equation of state of the full BFSS model we will use the same approach as for the bosonic matrix model and find mixed Gaussian states of fixed energy which maximize the von Neumann entropy. To this end we again split the coordinate and momentum dispersions into the classical and quantum contributions, as in

Correspondingly, the fermionic contribution

Since fermions are assumed to be in the ground state at fixed

In order to maximize the von Neumann entropy

We now obtain the equation of state for the full BFSS model within the Gaussian state approximation by maximizing the von Neumann entropy

The ground-state energy of the BFSS model is smaller than that of the bosonic matrix model.

The coordinate dispersion

At high temperatures both energy and

Having obtained the equation of state and

As a side remark, let us note that the temperature-dependent energies, coordinate dispersions, and entropies obtained within the Gaussian state approximation satisfy the so-called Bekenstein bound

In this work we numerically solve Eqs.

We average simulation results over several (typically, between five and seven) random initial conditions as previously discussed in Sec.

To have a first look at the real-time dynamics described by Eqs.

In order to make a meaningful comparison of simulations with characteristic timescales which differ by several orders of magnitude, in Fig.

In Fig.

The decrease of the classical contribution and the corresponding increase of the quantum contribution become particularly large at high temperatures, which indicates a rapid spread of wave functions in configuration space driven by the chaotic dynamics of their centers. In this way the system approaches a state of dynamical equilibrium. However, despite this rearrangement, it turns out that the overall coordinate and momentum dispersions which determine the von Neumann entropy

Of course, one should keep in mind that the Gaussian state approximation probably becomes invalid at late times, more precisely, at a time of the order of several classical Lyapunov times

The late-time expectation value of the coordinate dispersion

In classical mechanics the distance between two infinitely close solutions of the equations of motions can be expressed in terms of the Poisson bracket

In order to treat the out-of-time-order correlator

In order to calculate the Lyapunov distance as a function of time, we use the first equation in

Time dependence of the Lyapunov distances

While the classical dynamics is always chaotic and exhibits a well-defined exponential growth of the Lyapunov distance, quantum corrections from the bosonic sector make the dynamics completely nonchaotic at sufficiently low temperatures, with Lyapunov distances which do not exhibit any growth, but rather oscillate with a characteristic frequency

In contrast, for the BFSS model Lyapunov exponents remain finite down to the lowest temperature that we consider, and Lyapunov distances exhibit a clear growth. At temperatures up to

For higher temperatures Lyapunov distances for the bosonic matrix model and the full BFSS model behave in very similar ways. At early times

We also note that at sufficiently early times, when the exponential growth of Lyapunov distances has not yet fully developed,

To summarize, there are no indications that in either the bosonic matrix model or the BFSS model do quantum corrections lead to faster growth of the Lyapunov distance than in the classical matrix mechanics. At least at sufficiently short evolution times this observation should be qualitatively accurate. To quantify all these observations, we extract the leading quantum Lyapunov exponents

Leading Lyapunov exponent as a function of temperature for the classical and quantum dynamics of the bosonic matrix model and the full BFSS model for

The temperature dependence

In the full BFSS model the effect of fermions is to remove the low-temperature confinementlike regime, so that the system remains in the deconfinement phase all the way down to zero temperature independently of gauging

Quantum entanglement between different d.o.f. in an interacting system provides a quantitative picture of the “scrambling” and spreading of quantum information. Entanglement can be quantified in terms of the entanglement entropy

Strictly speaking, in gauge theories (of which the BFSS model is a descendant) the splitting of the physical Hilbert space into a direct product

Numerical calculation of the entanglement entropy is typically a rather nontrivial task, especially for real-time evolution of interacting systems. Since the Gaussian state approximation which we use in this paper evolves pure states into pure states (see Appendix

We now use the prescription sketched above to calculate entanglement entropy for time-dependent correlators

The splitting of matrix entries of

We calculate the entanglement entropy separately for each random initial condition, that is, separately for each pure Gaussian state

Splitting of the matrix entries of

Time dependence of the entanglement entropy for different partitions of the Hilbert space in the bosonic matrix model (left) and in the full BFSS model (right) with

We observe that the entanglement entropy indeed exhibits an expected universal “scrambling” behavior: a roughly linear growth at early times and saturation at late times. Only for the low-temperature regime of the bosonic matrix model does the growth appear to be so slow that we do not see the onset of saturation up to

For the bosonic matrix model the saturation value of the entanglement entropy per d.o.f.

The observation that

Approximately linear scaling of entanglement entropy with the number of d.o.f.

When the number of d.o.f. becomes comparable with the maximal value

In the full BFSS model (plots on the right side of Fig.

In order to quantify the timescale for the saturation of the entanglement entropy more precisely, we define the entanglement saturation time

By analogy with Lyapunov exponents, we introduce the inverse entanglement saturation time

Inverse entanglement saturation time

We also find that in the high-temperature regime for both the bosonic matrix model and the full BFSS model the entanglement saturation time

As we approach the low-temperature regime, the entanglement saturation time for the bosonic matrix model quickly decreases and becomes significantly smaller than the classical Lyapunov exponent, in agreement with the nonchaotic nature of the low-temperature regime. On the other hand, for the full BFSS model the temperature dependence of

Finally, in Fig.

A comparison of the temperature dependence of the leading Lyapunov exponent

While Lyapunov exponents and entanglement generation define the characteristic times for the onset of chaos and spreading of quantum information, the diffusion-driven approach to thermal equilibrium is characterized by another timescale

In our numerical study of Lyapunov distances we drive the system out of the dynamical equilibrium state by introducing a small coordinate shift

Signatures of “quasinormal ringing” in the early-time evolution of the Lyapunov distances

Nevertheless, it is interesting to study the temperature behavior of quasinormal frequencies at least at the qualitative level. To this end we first identify the duration of the ringing as the period of time during which each successive maximum of the Lyapunov distance

We show our estimates for the real and imaginary parts of the quasinormal frequency

Temperature dependence of the real (on the left) and imaginary (on the right) parts of the quasinormal frequency

The temperature dependence of the real part of

For the full BFSS model the temperature dependence of

The imaginary part of

In contrast, for the full BFSS model imaginary parts of quasinormal frequencies are much larger and seem to remain finite even at the lowest temperatures that we consider, which is again in agreement with the absence of a confinement regime in the BFSS model. While at very high temperatures we cannot reliably extract

Finally, let us note that at high temperatures we can use Eqs.

Until recently the real-time dynamics of the BFSS model could only be addressed either in the low-energy regime, which is tractable in terms of the dual holographic description, or in the high-energy regime in which the system becomes effectively classical. Our simulations of the corresponding ungauged models

The Gaussian state approximation appears to be much more accurate for the bosonic matrix model than for the BFSS model, as suggested by the quantitatively good agreement of our equation of state

We have explicitly studied and confirmed some of the important features of the real-time dynamics of the bosonic matrix model and the BFSS model which fit expectations either based on general grounds

Quantum real-time dynamics is characterized by smaller Lyapunov exponents in comparison with the classical system at the same energy. This ensures the validity of the MSS bound

The gauged bosonic matrix model becomes confining and nonchaotic at low temperatures

When fermionic d.o.f. are added, the BFSS model exhibits chaotic behavior and fast decay of quasinormal modes at all temperatures, in agreement with the absence of a confining regime all the way down to zero temperature

Entanglement entropy shows the expected scrambling behavior

The characteristic time

Our results for the time dependence of the entanglement entropy are under the best theoretical control, as they are extracted from the early-time behavior for which the Gaussian state approximation should be quantitatively accurate. At the same time, the relatively smooth time dependence allows for a rather unambiguous definition of the entanglement saturation time

Our real-time analysis might be further improved if one considers correlators with more than two canonical variables, e.g., within the

In fact, we have already tried to extend Eqs.

It would also be interesting to understand the applicability of the Gaussian state approximation to real-time dynamics of higher-dimensional gauge theories, as well as its interpretation in the context of the conventional scale separation between soft-momenta and hard-momenta gauge fields which allows one to treat the dynamics of soft gauge fields classically. In principle, Gaussian state approximation should extend the range of validity of real-time simulations of the Yang-Mills theory beyond that of the classical dynamics at a numerical cost which scales quadratically with spatial volume, which is thus comparable with the numerical cost of real-time simulations with fermions

P. B. is supported by the Heisenberg Fellowship from the German Research Foundation, Project No. BU2626/3-1. M. H. acknowledges JSPS KAKENHI Grant No. 17K14285. M. H. thanks the Erwin Schrödinger International Institute for Mathematics and Physics for the hospitality during the workshop “Matrix Models for Noncommutative Geometry and String Theory.” The authors also acknowledge hospitality of the ECT* Workshop “Quantum Gravity meets Lattice QFT” where part of this work was done. This work was also partially supported by the Department of Energy, Award No. DE-SC0017905. The calculations were performed on the “iDataCool” cluster at Regensburg University and on the LRZ cluster in Garching. The authors acknowledge valuable discussions with D. Berenstein, N. Bodendorfer, D. O’Connor, P. Romatschke, and A. Rothkopf.

Equations of motion

Here we give explicit expressions for these conserved quantities:

The conservation of the supersymmetry generators

To understand the fate of supersymmetry in more detail, let us first present an outline of the proof of the conservation of supersymmetry generators

Now we turn to the Gaussian state approximation and take the limit in which the classical expectation values

It is now easy to see that due to these brackets, one can no longer cyclically permute different

In our approximation the thermodynamics and the real-time evolution of Majorana fermions

In order to describe the ground state of

It is easy to see that the eigenvectors which correspond to energies

Decomposing the operators

Having fixed the initial conditions for the fermionic correlators

This way of solving Eq.

As discussed in Sec.

To begin with, we introduce the condensed index notation

It is now convenient to introduce the symmetric and real matrix

In order to solve Eqs.

After that we update the bosonic correlators

Finally, the variables

The commutator representation

All the data presented in the main text of the paper were obtained with

We use the following explicit form of the

To demonstrate the conservation of the angular momentum

For the proof of the conservation of supersymmetry generators