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The complete $1/N$ expansion of a SYK--like tensor model

Razvan Gurau

TL;DR

The paper analyzes the complete $1/n$ expansion of a SYK-like tensor model with $D+1$ colors and tensor fermions, showing the leading two-point function is melonic and the leading four-point functions are dressed ladders built from $G_{\rm LO}$. A key result is that any fixed order in $1/n$ can be expressed as a finite sum of convolutions of $G_{\rm LO}$ with a finite set of LO four-point kernels, avoiding the need to resum the full parquet family. The authors introduce core and reduced-scheme techniques to classify graphs by degree and prove finiteness of reduced schemes at fixed degree, ensuring a controllable, nonperturbative structure for all orders in $1/n$. These findings also clarify the nonperturbative content of OTOC-related ladders and provide a robust framework to extend analyses to arbitrary $2p$-point functions. Overall, the work delivers a rigorous, scalable blueprint for the complete $1/n$ expansion in SYK-like tensor models and clarifies how nonperturbative effects arise solely from ladder resummations anchored by LO data.

Abstract

A SYK--like model close to the colored tensor models has recently been proposed \cite{Witten:2016iux}. Building on results obtained in tensor models \cite{GurSch}, we discuss the complete $1/N$ expansion of the model. We detail the two and four point functions at leading order. The leading order two point function is a sum over melonic graphs, and the leading order relevant four point functions are sums over dressed ladder diagrams. We then show that any order in the $1/N$ series of the two point function can be written solely in term of the leading order two and four point functions. The full $1/N$ expansion of arbitrary correlations can be obtained by similar methods.

The complete $1/N$ expansion of a SYK--like tensor model

TL;DR

The paper analyzes the complete expansion of a SYK-like tensor model with colors and tensor fermions, showing the leading two-point function is melonic and the leading four-point functions are dressed ladders built from . A key result is that any fixed order in can be expressed as a finite sum of convolutions of with a finite set of LO four-point kernels, avoiding the need to resum the full parquet family. The authors introduce core and reduced-scheme techniques to classify graphs by degree and prove finiteness of reduced schemes at fixed degree, ensuring a controllable, nonperturbative structure for all orders in . These findings also clarify the nonperturbative content of OTOC-related ladders and provide a robust framework to extend analyses to arbitrary -point functions. Overall, the work delivers a rigorous, scalable blueprint for the complete expansion in SYK-like tensor models and clarifies how nonperturbative effects arise solely from ladder resummations anchored by LO data.

Abstract

A SYK--like model close to the colored tensor models has recently been proposed \cite{Witten:2016iux}. Building on results obtained in tensor models \cite{GurSch}, we discuss the complete expansion of the model. We detail the two and four point functions at leading order. The leading order two point function is a sum over melonic graphs, and the leading order relevant four point functions are sums over dressed ladder diagrams. We then show that any order in the series of the two point function can be written solely in term of the leading order two and four point functions. The full expansion of arbitrary correlations can be obtained by similar methods.

Paper Structure

This paper contains 18 sections, 3 theorems, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discussion
  3. Four point functions.
  4. The full $1/n$ series and non perturbative effects.
  5. Two and four point functions
  6. The two point function
  7. The four point functions
  8. The leading order in $1/n$
  9. First definition of two point melonic graphs critical.
  10. Second definition of two point melonic graphs GurSch.
  11. Closed melonic graphs.
  12. Four point chain graphs.
  13. Two point function at leading order
  14. Four point functions at leading order
  15. The complete $1/n$ series of the two point function
  16. ...and 3 more sections

Key Result

Proposition 1

critical For $D\ge 3$ a closed graph has degree zero if and only if it is a closed melonic graph.

Figures (4)

  • Figure 1: Closed, bipartite, edge $3+1$ colored, rooted graphs. The root is represented as dashed.
  • Figure 4: First definition of two point melonic graphs.
  • Figure 5: Second definition of two point melonic graphs.
  • Figure 6: A dipole, an unbroken and a broken chain.

Theorems & Definitions (3)

  • Proposition 1
  • Theorem 1
  • Theorem 2