Lyapunov Exponent Analysis Enables Prediction of Chaos in High-Dimensional Quantum Systems

The fundamental connection between classical chaos and its quantum counterpart remains a central question in physics, and recent work by Fabian Haneder, Gerrit Caspari, Juan Diego Urbina, and Klaus Richter from the Institut für Theoretische Physik, Universität Regensburg, sheds new light on this relationship. The team investigates how the rate of information scrambling, a key characteristic of chaotic systems, emerges in quantum systems that possess a classically chaotic counterpart, focusing on the growth rate of Out-of-Time-Ordered Commutators as a diagnostic tool. Their research demonstrates a clear link between the classical Lyapunov exponent, a measure of chaoticity, and the quantum scrambling rate, revealing a universal behaviour as the system’s complexity increases. Significantly, the team finds evidence of maximally fast scrambling within a well-defined classical Hamiltonian system, without relying on external factors like disorder, and provides further insight into the mechanisms underlying the celebrated Maldacena-Shenker-Stanford bound on chaos, bolstering the connection between quantum chaos and two-dimensional gravity.

Universal Quantum Chaos Scaling in Hyperbolic Systems

Scientists demonstrate a consistent semiclassical theory to calculate quantum Lyapunov exponents in systems with a large number of degrees of freedom, establishing a well-defined classical limit for these calculations. The research focuses on out-of-time-ordered commutators (OTOCs), key indicators of scrambling, a facet of short-time chaos, and applies this approach to quantized high-dimensional hyperbolic motion, a chaotic system exhibiting gravity-like correlations. Measurements of the OTOC growth rate as a function of the number of degrees of freedom, n, and inverse temperature, β, reveal a scaled growth rate describable by a universal function of n. This function displays a crossover from classical to quantum behaviour as n increases and/or temperature decreases.

Experiments demonstrate that in the limit of infinite n, the system exhibits maximally fast scrambling, saturating the Maldacena-Shenker-Stanford (MSS) bound on chaos. The team elucidates the non-perturbative mechanism underlying this saturation, identifying contributions to the mean density of states as critical. Data confirms that this saturation is a quantum effect stemming from quantum corrections to the leading power law behavior of the density of states, providing explicit results for the quantum Lyapunov exponent as a function of both temperature and the number of degrees of freedom. The team rigorously shows the saturation of the MSS bound, demonstrating that the growth rate, Λ, obeys the inequality Λ ≤ 2π ħβ.

Measurements confirm that the system is a fast scrambler, and in the correct limit, a maximally fast scrambler, as expected for a system dual to two-dimensional gravity. Further analysis takes initial steps toward evaluating the first subleading ħ² correction to the OTOC, which is expected to exhibit exponential behavior with a different, likewise bounded growth exponent. These results provide the first evidence of maximally fast scrambling in a chaotic system with a well-defined classical Hamiltonian limit, without invoking external mechanisms like disorder averaging.

OTOC Growth Rate Reveals Quantum Scrambling Transition

This research presents a detailed investigation into quantum chaos and scrambling, focusing on the out-of-time ordered commutator (OTOC) as a key diagnostic tool. Scientists have established a consistent approach to calculating the growth rate of the OTOC for systems exhibiting chaotic behaviour, linking it to both the classical Lyapunov exponent and the density of states. Applying this method to a specifically defined chaotic system, a particle moving on a high-dimensional hyperbolic manifold, the team computed the OTOC growth rate as a function of the number of degrees of freedom and temperature. The results demonstrate a transition from classical to quantum behaviour as the number of degrees of freedom increases or the temperature decreases, ultimately revealing maximally fast scrambling in the limit of infinite degrees of freedom. This finding supports the connection between this dynamical system and two-dimensional gravity, and elucidates the mechanism by which the bound on chaos is saturated through contributions to the mean density of states. The authors acknowledge that their calculations rely on approximations and that further investigation is needed to fully understand the behaviour of the system in all regimes.

Quantum Chaos on Curved Manifolds Explained

Scientists have established a framework to connect classical and quantum descriptions of systems on curved manifolds. This involves analyzing the terms in an expansion of the Hamiltonian to determine when classical physics provides an accurate approximation, and deriving a condition relating system parameters to ensure this classical regime is valid. The Weyl symbol of the Hamiltonian, a way to represent the quantum operator in classical phase space, is calculated using the DeWitt superoperator formalism. The research demonstrates that the quantum correction to the Weyl symbol is proportional to the Ricci scalar, a measure of the manifold’s curvature, indicating that the curvature of space affects quantum corrections to classical behaviour. Key concepts include the Hamiltonian, Weyl symbol, DeWitt superoperator, Ricci scalar, and Planck’s constant. This work provides a mathematical foundation for understanding how quantum effects modify classical behaviour in systems defined on curved space.