Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models (1609.07832v2)

Abstract: The Sachdev-Ye-Kitaev model is a $(0+1)$-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with $N$ Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large $N$ limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a "butterfly velocity" describing the propagation of chaos in space. We mainly present results for a $(1+1)$-dimensional example, and discuss the general case near the end.

• The paper introduces a generalized SYK model on lattices, extending the classic (0+1)-dimensional system to capture local criticality and chaotic behavior.
• The study employs analytical techniques to reveal diffusive energy transport, quantified by a distinct diffusion constant and butterfly velocity.
• The findings provide fresh insights into holographic principles and quantum chaos in strongly correlated systems, paving the way for future explorations.

Overview of "Local Criticality, Diffusion and Chaos in Generalized Sachdev-Ye-Kitaev Models"

The paper "Local Criticality, Diffusion and Chaos in Generalized Sachdev-Ye-Kitaev Models" by Yingfei Gu, Xiao-Liang Qi, and Douglas Stanford proposes a fascinating extension of the Sachdev-Ye-Kitaev (SYK) model to higher dimensions. The SYK model, originally a (0+1)-dimensional system, is notable for its connections to quantum chaos and holography, despite its limited dimensionality. The authors tackle this limitation by designing a generalized variant capable of being defined on any lattice in arbitrary dimensions, preserving key SYK properties while introducing new phenomena such as diffusive energy transport and spatial chaos propagation.

Key Features and Results

1. Generalization to Higher Dimensions: The proposed models expand upon the SYK framework, applying it to a lattice with $N$ Majorana fermions at each site and incorporating random interactions both within and between sites. The generalization aims to explore the range of phenomena possible in systems that extend beyond the microcosmic scale of the traditional SYK model.
2. Preservation of SYK Characteristics: In the large $N$ limit, the generalized higher-dimensional models retain several pivotal SYK characteristics. Specifically, they exhibit local criticality in two-point functions, non-zero entropy at zero temperature, and chaotic behavior as measured by out-of-time-ordered correlators (OTOCs).
3. New Phenomena in Higher Dimensions: By transitioning to a lattice framework, the authors introduce the ability to paper energy transport. They identify a haLLMark of the generalized models as diffusive energy transport, characterized by a diffusion constant. Furthermore, they define the "butterfly velocity" as a measure of the spatial propagation of chaos, echoing phenomena observed in holographic systems.
4. Specific Example in Two Dimensions: The paper provides a detailed analysis of a generalized (1+1)-dimensional SYK lattice model. This example serves as a concrete demonstration of the model's adaptability and the implications of what happens when boosting the SYK model to two spatial dimensions.

Implications and Speculations

The ability to generalize the SYK model to higher dimensions with preserved characteristics and new features carries significant implications for both theoretical and practical fields:

• Theoretical Insights: These models offer a novel playground for exploring holographic principles in higher dimensions, potentially providing new insights into the AdS/CFT correspondence beyond the original (0+1)-dimensional confines. The work hints at a potential duality between such generalized SYK models and gravitational theories involving incoherent black holes—objects that exhibit similar diffusion and chaos characteristics.
• Modeling Quantum Chaos: The constructed framework provides a new method to paper chaotic dynamics within more realistic physical systems that extend into higher dimensions. The generalized SYK model could serve as a potent tool in unraveling complexity in strongly correlated electron systems, particularly those that display non-Fermi liquid behavior.
• Potential Extensions: While the paper primarily paves the way for understanding diffusion and chaos in higher dimensions, future studies could probe additional dynamical characteristics, such as thermalization and the nature of many-body localization at varied parameter regimes.

Conclusion

The work presented by Gu, Qi, and Stanford represents a significant stride in extending the analytical power of the SYK model to embrace higher-dimensional complexities. By building on the bad metals and AdS/CFT connection theories, they propose a model that maintains the analytical tractability of the SYK model while embracing broader spatial and temporal dynamics. This extension not only bridges the gap between low and higher-dimensional quantum models but also lays the groundwork for future explorations into how these dimensions can reflect broader cosmic and theoretical phenomena. The prospects are inviting, urging questions about the reach of holography, local criticality, and chaos in yet-to-be-discovered states of matter.