Entanglement spreading in a minimal model of maximal many-body quantum chaos (1812.05090v3)

Abstract: The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin chains. Specifically, we consider the so called "self-dual" kicked Ising chains initialised in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the models considered are maximally chaotic: their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so called "minimal cut" bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel "duality-based" numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is non-integrable.

Entanglement Spreading in Maximally Chaotic Spin Chains

The paper "Entanglement spreading in a minimal model of maximal many-body quantum chaos" by Bruno Bertini, Pavel Kos, and Tomaž Prosen presents a rigorous mathematical framework for analyzing entanglement dynamics in out-of-equilibrium quantum spin chains that are "maximally chaotic." The authors focus on self-dual kicked Ising chains, which transition between integrability and ergodic behavior through adjustments in the longitudinal magnetic field.

Overview of Key Contributions

This paper introduces an analytical method for computing the entanglement dynamics in these spin chains, offering insights into entanglement spreading and its implications for both integrable and non-integrable systems. Key contributions include:

1. Exact Results for Entanglement Dynamics: The paper provides exact calculations for the time evolution of entanglement entropies, showcasing that these results are independent of the field configuration at self-dual points. Such precision is rare in non-integrable systems.
2. Entanglement Entropy Saturation: The entanglement entropies exhibit a universal saturation behavior, described by $S^{(\alpha)}_A(t) = \min(2t, N) \log 2$, where $N$ is the subsystem size, independent of the specifics of the magnetic fields or the chaoticity of the underlying dynamics.
3. Modeling Non-Integrable Behavior with Integrable Dynamics: Even though the self-dual kicked Ising chain is non-integrable for a wide parameter space, the entanglement dynamics can be exactly modeled, revealing universal characteristics in how entanglement grows and saturates across different setups.
4. Implications for Hamiltonian Ergodicity: The authors demonstrate that the ergodicity in these quantum systems leads to a saturation of entropies at values consistent with thermalization predictions. However, integrable systems show a nuanced dependency on initial conditions concerning saturation values.
5. Duality-Based Numerical Approaches: The work extends the analytical framework to numerical studies by implementing a duality method for entanglement entropies, providing evidence that the exact results describe the leading-order behavior for generic product states when the model is non-integrable.

Implications and Future Directions

The findings have important implications for understanding complex quantum systems' dynamics, specifically in contexts where non-integrable dynamics are prevalent, such as thermalization, many-body localization, and quantum transport processes. The techniques developed herein may serve as foundational tools to paper quantum chaos and entanglement spreading in a broader set of quantum many-body systems.

Future work stemming from this paper could explore:

• Generalizing to Other Models: Investigating whether similar mathematical structures and duality insights can be applied to other models of quantum chaos, potentially leading to further exact results.
• Disjoint Block Entanglement: As suggested by the authors, exploring entanglement across disjoint blocks might reveal differences in the dynamics for integrable versus non-integrable systems.
• Perturbation Theory Approaches: Analyzing perturbations around the self-dual points to see how these affect entanglement dynamics and whether they lead to qualitatively different quantum behaviors.

The framework and results described enrich our understanding of entanglement in chaotic and complex quantum systems, providing a robust platform for future research in quantum information and many-body dynamics.