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 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.
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 two classes of quantum phenomena associated with classical chaos in a variety of quantum models: (i) dynamical localization and its extension and generalization to interacting few- and many-body systems and (ii) quantum exponential divergences in high-order correlators and other diagnostics of quantum chaos. Dynamical localization (DL) is a subtle phenomenon related to Anderson localization. It hinges on quantum interference and is typically destroyed in presence of interactions. DL often manifests as a failure of a driven system to heat up, violating the foundations of statistical physics. Kicked rotor (KR) is a prototypical chaotic classical model that exhibits linear energy growth with time. The quantum kicked rotor (QKR) features DL instead: its energy saturates. Multiple attempts of many-body generalizations faced difficulties in preserving DL. Recently, DL was shown in a special integrable many-body model. We study non-integrable models of few- and many-body QKR-like systems and provide direct evidence that DL can persist there. In addition, we show how a novel related concept of localization landscape can be applied to study transport in rippled channels. Out-of-time-ordered correlator (OTOC) was proposed as an indicator of quantum chaos, since in the semiclassical limit, this correlator's possible exponential growth rate (CGR) resembles the classical Lyapunov exponent (LE). We show that the CGR in QKR is related, but distinct from the LE in KR. We also show a singularity in the OTOC at the Ehrenfest time tᴱ due to a delay in the onset of quantum interference. Next, we study scaling of OTOC beyond tᴱ. We then explore how the OTOC-based approach to quantum chaos relates to the random-matrix-theoretical description by introducing an operator we dub the Lyapunovian. Its level statistics is calculated for quantum stadium billiard, a seminal model of quantum chaos, and aligns perfectly with the Wigner-Dyson surmise. In the semiclassical limit, the Lyapunovian reduces to the matrix of uncorrelated finite-time Lyapunov exponents, connecting the CGR at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference. Finally, we consider quantum polygonal billiards: their classical counterparts are non-chaotic. We show exponential growth of the OTOCs in these systems, sharply contrasted with the classical behavior even before quantum interference develops.
Abstract We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.
The concept of out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further we demonstrate an equivalent formalism of computation using a general time independent Hamiltonian having well defined eigenstate representation for integrable supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism we consider the two well known models viz. Harmonic Oscillator and one dimensional potential well within the framework of supersymmetry. For the Harmonic Oscillator case we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both microcanonical and canonical ensembles in quantum mechanics without supersymmetry. On the other hand, for one dimensional potential well problem we found significantly different time scale and the other parameter dependence compared to the results obtained from non-supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model independent Hamiltonian along with the previously mentioned these well cited models.
A bstract We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos . Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Rényi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the “weakly chaotic” character of these eigenstates. Owing to their simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.