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SYK-like Tensor Models on the Lattice

Prithvi Narayan, Junggi Yoon

TL;DR

<3-5 sentence high-level summary>This paper develops a lattice realization of SYK-like tensor models, focusing on KT chain, GW, and generalized GW constructions, and extends to rank-D variants. By summing melonic two-point diagrams and ladder four-point diagrams across Cooper, Pillow, and Tetrahedron channels, it derives kernel structures, spectra, and chaos properties at large N and strong coupling, with explicit expressions for diffusion and Lyapunov exponents. A key finding is that Cooper-channel dynamics saturate the chaos bound (λ_L = 2π/β) in KT and GW variants, while Pillow channels remain non-chaotic at leading order and reveal subleading chaotic behavior and a rich operator spectrum. The analysis generalizes to rank-D tensor models, showing analogous chaotic behavior and identifying next-to-subleading 1/N corrections for D>5, consistent with an emergent Schwarzian-like effective action. These results clarify how lattice structure and color contractions shape chaotic and transport properties in SYK-like tensor systems and open avenues for further rank-D and non-melonic explorations.

Abstract

We study large $N$ tensor models on the lattice without disorder. We introduce techniques which can be applied to a wide class of models, and illustrate it by studying some specific rank-3 tensor models. In particular, we study Klebanov-Tarnopolsky model on lattice, Gurau-Witten model (by treating it as a tensor model on four sites) and also a new model which interpolates between these two models. In each model, we evaluate various four point functions at large $N$ and strong coupling, and discuss their spectrum and long time behaviors. We find similarities as well as differences from SYK model. We also generalize our analysis to rank-$D$ tensor models where we obtain analogous results as $D=3$ case for the four point functions which we computed. For $D>5$, we are able to compute the next-to-subleading ${1 \over N}$ corrections for a specific four point function.

SYK-like Tensor Models on the Lattice

TL;DR

<3-5 sentence high-level summary>This paper develops a lattice realization of SYK-like tensor models, focusing on KT chain, GW, and generalized GW constructions, and extends to rank-D variants. By summing melonic two-point diagrams and ladder four-point diagrams across Cooper, Pillow, and Tetrahedron channels, it derives kernel structures, spectra, and chaos properties at large N and strong coupling, with explicit expressions for diffusion and Lyapunov exponents. A key finding is that Cooper-channel dynamics saturate the chaos bound (λ_L = 2π/β) in KT and GW variants, while Pillow channels remain non-chaotic at leading order and reveal subleading chaotic behavior and a rich operator spectrum. The analysis generalizes to rank-D tensor models, showing analogous chaotic behavior and identifying next-to-subleading 1/N corrections for D>5, consistent with an emergent Schwarzian-like effective action. These results clarify how lattice structure and color contractions shape chaotic and transport properties in SYK-like tensor systems and open avenues for further rank-D and non-melonic explorations.

Abstract

We study large tensor models on the lattice without disorder. We introduce techniques which can be applied to a wide class of models, and illustrate it by studying some specific rank-3 tensor models. In particular, we study Klebanov-Tarnopolsky model on lattice, Gurau-Witten model (by treating it as a tensor model on four sites) and also a new model which interpolates between these two models. In each model, we evaluate various four point functions at large and strong coupling, and discuss their spectrum and long time behaviors. We find similarities as well as differences from SYK model. We also generalize our analysis to rank- tensor models where we obtain analogous results as case for the four point functions which we computed. For , we are able to compute the next-to-subleading corrections for a specific four point function.

Paper Structure

This paper contains 30 sections, 122 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Review
  3. Klebanov-Tarnopolsky Model
  4. Two point Function:
  5. Four point Function:
  6. Chaos in KT Model and the Lyapunov exponent:
  7. SYK model and Gu-Qi-Stanford Generalization Thereof
  8. Chaos in Gu-Qi-Stanford generalization of SYK model:
  9. Klebanov-Tarnopolsky Chain Model
  10. Two point function:
  11. Four Point Function in Klebanov-Tarnopolsky Chain Model
  12. Cooper Channel
  13. Pillow Channel
  14. Tetrahedron Channel
  15. Generalized Tensor Models on a Lattice
  16. ...and 15 more sections

Figures (27)

  • Figure 1: Cooper, Pillow and Tetrahedron contractions. Each vertex represents a fermion, and the colored edge denotes the contraction of gauge index of the corresponding color in the two fermions.
  • Figure 2: Free propagator and vertex in the KT model
  • Figure 3: Example : Melonic and non-melonic vacuum diagrams
  • Figure 6: Contributions to two point function in single line notation
  • Figure 7: Schiwinger-Dyson equation for two point function: strand & single line
  • ...and 22 more figures