Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos (1805.00931v3)

Abstract: The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable $t$ in the thermodynamic limit. In particular, we rigorously prove RMT form factor for odd $t$, while we formulate a precise conjecture for even $t$. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in non-integrable systems.

• The paper introduces a minimal periodically driven Ising model to reveal quantum chaos via exact spectral form factors.
• The study maps the spectral form factor to the classical partition function of the 2D Ising model, aligning analytical results with RMT predictions.
• The paper rigorously excludes many-body localization by showing that even minor disorder preserves ergodicity in the thermodynamic limit.

Analysis of the Exact Spectral Form Factor in Many-Body Quantum Chaos

The paper, "Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos," by Bertini, Kos, and Prosen presents a significant contribution to the field of many-body quantum chaos by investigating the application of Random Matrix Theory (RMT) to a quantum many-body system. The paper introduces a specific periodically driven Ising model that serves as a minimal model to showcase the chaos phenomena in the quantum domain without relying on semiclassical concepts, traditionally thought necessary to bridge quantum chaos with classical chaos.

Key Features of the Study

The authors explore the spectral correlations of energy levels in quantum systems, which are predominantly described by RMT. The model utilized in this research is the Floquet dynamics of a periodically driven Ising spin chain, characterized by both transverse and longitudinal fields. The spectral form factor (SFF), the primary analytical tool in this context, is linked to a classical partition function of the two-dimensional Ising model. This mapping illustrates the elegance of methodologies that relate quantum characteristics to classical statistical models.

Numerical and Analytical Results

The paper sets out two main accomplishments:

1. Ergodic Behavior in Spectral Statistics: The research finds profound agreements between the SFF of the quantum system at hand and the predictions made by RMT, particularly within the thermodynamic limit. Notably, for odd time variables, the analytical results fully coincide with RMT. The authors extrapolate the results to formulate a mathematical conjecture for even time intervals, presenting a consistent picture of quantum ergodicity.
2. Exclusion of Many-Body Localization: By introducing disorder in the longitudinal field and considering its effects in the thermodynamic limit, the manuscript successfully rigorously excludes the possibility of many-body localization within this model. The result is crucial, suggesting that even small disorder induces ergodicity, negating localization in this driven system.

Theoretical and Practical Implications

The implications of this work are profound. It provides a clearer route to understanding the intrinsic quantum nature of chaos in many-body systems and sets a framework for further exploration of RMT in non-integrable systems. The adoption of a clean spin model devoid of semiclassical limits presents exciting opportunities for scholars aiming to deepen the theoretical foundations of quantum chaos.

Speculations on Future Developments

Moving forward, applying the techniques and results of this paper could offer potent insights into Entanglement Entropy dynamics and the Eigenstate Thermalization Hypothesis (ETH) in similar quantum systems. Additionally, the methodology presented could be adapted to evaluate dynamical correlation functions in complex quantum settings, potentially unraveling new insights into decoherence and thermalization processes in quantum systems.

In conclusion, this paper effectively bridges a crucial gap in our comprehension of quantum chaos, providing a compelling framework for understanding how random matrix theory emerges in quantum many-body systems with no classical correspondence. By addressing these dynamics using exact methods within the thermodynamic limit, the research paves the way for a deeper understanding of chaotic behavior in complex quantum systems.