All point correlation functions in SYK (1710.08113v2)

Abstract: Large $N$ melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary $O(N)$ invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in $1/N$, through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for $q$-body SYK, at any $q$. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.

• The paper presents the complete solution for connected fermion $2p$-point correlation functions in the SYK model at leading \(1/N\) order, showing the six-point function underlies higher orders.
• The authors leverage the SYK model's conformal symmetry to formulate correlation functions using conformal blocks and propose Feynman-like rules for computing higher-point functions.
• The study proposes a universal structure for correlation functions of large dimension operators and simplifies computations using contour integrals, demonstrating analytic tractability.

Overview of "All point correlation functions in SYK"

The paper by Gross and Rosenhaus explores the understanding of correlation functions within the Sachdev-Ye-Kitaev (SYK) model, particularly focusing on exploiting the conformal structure emergent in the large-N limit of these models. Developing insights into correlation functions is crucial for discerning the nature of the quantum mechanical systems described by these models, which are believed to be holographically dual to black holes in two-dimensional anti-de Sitter space (AdS$_2$).

Key Contributions

The research highlights a methodical analysis of the SYK model through the presentation of the complete solution for the connected fermion $2p$-point correlation function. This is achieved within the leading nontrivial order in $1/N$, where $N$ denotes the number of Majorana fermions. Notably, the study emphasizes the six-point function, which is shown to underlie the entire structure of all higher-order correlation functions, thereby setting a pivotal groundwork for the architectural understanding of SYK correlation functions.

Several innovative aspects of this study include:

• Conformal Structure Utilization: The authors leverage the conformal symmetry of the SYK model at strong coupling to cement the formulation of all correlation functions in terms of conformal blocks. This methodological approach is a significant leap in rendering these complex calculations computationally feasible.
• Analytic Setup for Universality: The paper proposes that the correlation functions for operators with large dimension manifest a universal structure. These can be somewhat tractable using the method of a saddle-point analysis, emphasizing the large $N$ behavior that characterizes melonic diagrams.
• Feynman-like Rules: A set of rules analogous to Feynman diagrams is proposed to compute higher-point correlation functions. Notably, each higher-order correlation function essentially comes down to a sum over specific structured diagrams recognized from simpler graphs like ladders and contact diagrams.

Strong Numerical Results and Claims

Among the significant claims, the universal nature of certain high-dimension operator correlators is substantiated. The research provides a robust framework where computational insights blend seamlessly with analytic approaches. The reframing of various correlation functions as integrals over simpler dual graphs underpins a computational avenue that majors on the decomposition into contour integrals involving hypergeometric functions at critical understanding nodes.

One remarkable result is the simplification via contour integrals, which vouches for an elegance in traversing beyond naive dimensional analyses into methodological depth with saddle point solutions.

Implications and Future Directions

Theoretical Implications: From a theoretical standpoint, the dual basis for these operators sets the stage for comparative studies in higher-dimensional holographic correspondences, with potential implications on constructing dual theories resembling gravitational interactions in disordered systems.

Practical Applications: While the immediate practical applications are narrowly defined due to the abstract nature of the content, the formulation lays a solid foundation for analogue quantum computations and potential integration into machine learning frameworks where complex systems display inherently chaotic behavior.

Speculative Future Developments:

1. The compelling analogy between the SYK model's structural properties and quantum gravity in low-dimensional spaces could prompt further exploration into string theory aspects.
2. There is potential for using these findings to explore quantum chaos and its relation to thermalization processes within quantum systems.
3. Further investigation into the role of tensor models could lead to enriched understandings across other physical models exhibiting melonic dominance.

In summary, the paper provides a concrete methodological advancement in handling SYK correlation functions, marrying analytic techniques with numerical approaches to unveil deeper features of quantum systems at large scales. This work not only solidifies components of the SYK model's holographic correspondence but also opens routes for further theoretical explorations anchoring on symmetry exploits.