From Randomness to Order: Physicists Map a Chaos to Integrability Transition

Leon Miyahara and Shono Shibuya at Nagoya University show this crossover using a framework called BPS chaos, analysing the spectrum of an operator projected onto a specific subspace. Their numerical findings reveal random-matrix behaviour approaching the chaotic limit, smoothly shifting to Poisson statistics as the model nears integrability. The analysis offers a key example of diagnosing a chaos-integrability transition solely through the examination of BPS states, advancing understanding of complex quantum systems.

BPS state spectral statistics reveal a chaos-integrability transition in a supersymmetric SYK model

A gap ratio of 0.598, consistent with Gaussian Unitary Ensemble (GUE) predictions, has transitioned to values indicative of Poisson statistics within a supersymmetric model’s BPS subspace. It marks the first time a chaos-integrability transition has been diagnosed solely by examining these restricted BPS states, previously requiring analysis of non-BPS states. This simplification supports the investigation of quantum chaos. The analysis focuses on a deformed N=2 supersymmetric SYK model, interpolating between chaotic and integrable systems, and reveals a clear crossover in spectral statistics. The Sachdev-Yeom-Kitaev (SYK) model, a solvable model of quantum gravity, has become a crucial tool in understanding the emergence of spacetime and black hole physics. This research builds upon that foundation, exploring a supersymmetric extension, the N=2 SYK model, and deforming it to create a system that smoothly transitions between chaotic and integrable behaviour. The BPS subspace, a specific sector of the supersymmetric model preserving a certain amount of symmetry, is particularly well-suited for this investigation due to its simplified structure and reduced computational complexity.

The operator spectrum projected onto the BPS subspace revealed a clear shift in statistical behaviour. Initially, it exhibited characteristics consistent with random-matrix theory, before transitioning as the model approached an integrable state. This change was evidenced by a move towards Poisson statistics, indicating a loss of the spectral correlations associated with chaotic systems. Furthermore, the spectral form factor displayed a characteristic “slope-ramp-plateau” structure in the chaotic regime, which diminished as integrability was approached. The spectral form factor, a measure of the fluctuations in the energy levels, provides a sensitive probe of the underlying quantum chaos. In the chaotic regime, the spectral form factor typically exhibits a characteristic “slope-ramp-plateau” structure, reflecting the long-range correlations between energy levels. As the system approaches integrability, these correlations weaken, and the spectral form factor loses its distinctive shape. The observed transition in the spectral form factor provides insight into the long-range correlations within the system, highlighting how these correlations weaken as the system becomes more integrable. The GUE, a cornerstone of random-matrix theory, predicts the statistical distribution of energy levels in chaotic systems. The observed gap ratio of 0.598 aligns closely with GUE predictions, confirming the presence of chaotic behaviour in the SYK limit.

Identifying order-chaos transitions via BPS subspace spectral statistics

Understanding how order emerges from chaos is increasingly the focus of research, a question central to fields ranging from quantum gravity to the behaviour of complex materials. This analysis demonstrates a new method for tracking this transition, analysing a specific, restricted portion of a complex system, the BPS subspace, to reveal shifts between chaotic and orderly states. The ability to diagnose a chaos-integrability transition within a restricted subspace is significant because it simplifies the analysis and reduces computational demands. Traditionally, such investigations required analysing the full spectrum of the system, including non-BPS states, which can be computationally expensive. By focusing solely on the BPS subspace, the researchers have developed a more efficient and tractable approach. The authors acknowledge a significant limitation; their current model relies on an exponential dependence on system size, chosen to mimic other theoretical frameworks like the Rosenzweig-Porter model.

This reliance on an artificially scaled system size is important to consider, but the analysis offers a valuable new set of tools for investigating the transition between order and chaos, potentially extending to other systems. The Rosenzweig-Porter model, a widely studied model of quantum chaos, exhibits similar scaling behaviour, allowing for direct comparison and validation of the results. While the exponential scaling is a simplification, it allows the researchers to explore the essential physics of the chaos-integrability transition without being overwhelmed by computational complexity. Analysing how energy levels are distributed provides a clear pathway for identifying this shift using spectral statistics, and demonstrates the technique’s effectiveness. Spectral statistics, the study of the statistical properties of energy levels, provides a powerful tool for diagnosing quantum chaos. Different statistical distributions correspond to different types of quantum behaviour, allowing researchers to distinguish between chaotic, integrable, and intermediate regimes. This analysis establishes a new diagnostic tool for identifying transitions between chaotic and orderly behaviour in complex quantum systems.

A simplified region within a supersymmetric model, the BPS subspace, allowed scientists to demonstrate a clear shift in spectral statistics, moving from behaviour resembling the SYK model to characteristics of an integrable system. This focused approach streamlines investigations, opening questions regarding the universality of this technique across different physical systems and its potential application to diverse areas of physics. The universality of this technique remains an open question. While the analysis has been performed within the specific context of the N=2 supersymmetric SYK model, the underlying principles may apply to other systems exhibiting similar chaos-integrability transitions. Further research will explore whether this BPS subspace method can be applied to other models and physical scenarios. Potential applications include the study of many-body systems in condensed matter physics, quantum field theories, and even the dynamics of classical chaotic systems. Investigating the robustness of this method to variations in the model parameters and the inclusion of additional interactions will be crucial for establishing its broader applicability.

The research demonstrated a transition from chaotic to orderly behaviour within a simplified model of quantum mechanics. Spectral statistics, analysing the distribution of energy levels, revealed a shift from random-matrix behaviour, characteristic of the SYK model, to Poisson statistics, indicative of an integrable system. This provides a direct example of a chaos-integrability crossover identified using only a specific, restricted portion of the model known as the BPS subspace. The authors intend to explore whether this method can be applied to other physical systems exhibiting similar transitions.