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Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders

Valentin Bonzom, Luca Lionni, Adrian Tanasa

TL;DR

This work analyzes the diagrammatics of colored SYK and Gurau–Witten tensor models in the large-$N$ expansion, using edge-colored graphs and the contraction $G_{/0}$ to classify leading and subleading contributions to 2-point and 4-point functions. It shows melonic dominance at leading order and ladder-like chains at next-to-leading order for the colored SYK model, and extends the analysis to the Gurau–Witten model where non-bipartite graphs can appear, causing richer NLO/NNLO behavior. At LO (and NLO) the colored SYK and Gurau–Witten models coincide for 2-point functions, but their 4-point function diagrams can differ, with broken vs unbroken chains lifting degeneracies in the tensor model. The results establish a combinatorial framework for systematically extracting LO/NLO contributions and set the stage for applying these techniques to other SYK-like tensor models and to explicit amplitude computations.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large $N$ expansion, and argue that our method can be further applied if necessary. In a second part we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.

Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders

TL;DR

This work analyzes the diagrammatics of colored SYK and Gurau–Witten tensor models in the large- expansion, using edge-colored graphs and the contraction to classify leading and subleading contributions to 2-point and 4-point functions. It shows melonic dominance at leading order and ladder-like chains at next-to-leading order for the colored SYK model, and extends the analysis to the Gurau–Witten model where non-bipartite graphs can appear, causing richer NLO/NNLO behavior. At LO (and NLO) the colored SYK and Gurau–Witten models coincide for 2-point functions, but their 4-point function diagrams can differ, with broken vs unbroken chains lifting degeneracies in the tensor model. The results establish a combinatorial framework for systematically extracting LO/NLO contributions and set the stage for applying these techniques to other SYK-like tensor models and to explicit amplitude computations.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a model of interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large expansion, and argue that our method can be further applied if necessary. In a second part we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.

Paper Structure

This paper contains 21 sections, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalisations of SYK -- the colored SYK models
  3. Diagrammatics of the colored SYK model
  4. LO, NLO of vacuum and 2-point graphs
  5. Leading order
  6. Next-to-leading order and chains
  7. Following orders of the partition function
  8. LO and NLO of 4-point functions
  9. Diagrammatics of the Gurau-Witten model
  10. 2-point functions: LO, NLO and so on
  11. Results
  12. Proof
  13. 4-point function: LO, NLO, and so on
  14. Number of broken faces with two external colors.
  15. Number of broken faces with one external color.
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Melonic graphs of the SYK and colored SYK models
  • Figure 2: A melonic graph of the Gurau-Witten model.