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SYK Models and SYK-like Tensor Models with Global Symmetry

Junggi Yoon

TL;DR

The work analyzes SYK models with explicit global symmetry, deriving a bi-local collective action that exhibits an emergent ${\text{diff}} \ltimes \widehat{so}(M)$ symmetry at strong coupling and produces a Schwarzian plus particle-on-group low-energy effective action. By diagonalizing the quadratic fluctuations and computing four-point kernels, it identifies a rich spectrum with singlet, antisymmetric, and symmetric channels; the singlet sector saturates the chaos bound with Lyapunov exponent ${2\\pi}/{\beta}$, while the other channels show suppressed or linear growth. The tensor-model extension with $SO(M)$ symmetry mirrors the SYK results, with Cooper channels reproducing maximal chaos and Pillow channels generally non-exponentially growing at leading order, though subleading ladders for large $q$ can exhibit maximal growth, hinting at a Schwarzian-like mechanism. The paper also discusses a holographic gravity interpretation, including a 3D gravity conjecture for $q=4$, and suggests further exploration of higher-spin or $W$-type symmetries and their bulk duals. Overall, the work establishes a unified framework for SYK-type systems with global symmetries, revealing new symmetry structures, spectral towers, and potential gravity duals that deepen the connection between quantum chaos and holography.

Abstract

In this paper, we study an SYK model and an SYK-like tensor model with global symmetry. First, we study the large $N$ expansion of the bi-local collective action for the SYK model with manifest global symmetry. We show that the global symmetry is enhanced to a local symmetry at strong coupling limit, and the corresponding symmetry algebra is the Kac-Moody algebra. The emergent local symmetry together with the emergent reparametrization is spontaneously and explicit broken. This leads to a low energy effective action. We evaluate four point functions, and obtain spectrum of our model. We derive the low energy effective action and analyze the chaotic behavior of the four point functions. We also consider the recent 3D gravity conjecture for our model. We also introduce an SYK-like tensor model with global symmetry. We first study chaotic behavior of four point functions in various channels for the rank-3 case, and generalize this into a rank-$(q-1)$ tensor model.

SYK Models and SYK-like Tensor Models with Global Symmetry

TL;DR

The work analyzes SYK models with explicit global symmetry, deriving a bi-local collective action that exhibits an emergent symmetry at strong coupling and produces a Schwarzian plus particle-on-group low-energy effective action. By diagonalizing the quadratic fluctuations and computing four-point kernels, it identifies a rich spectrum with singlet, antisymmetric, and symmetric channels; the singlet sector saturates the chaos bound with Lyapunov exponent , while the other channels show suppressed or linear growth. The tensor-model extension with symmetry mirrors the SYK results, with Cooper channels reproducing maximal chaos and Pillow channels generally non-exponentially growing at leading order, though subleading ladders for large can exhibit maximal growth, hinting at a Schwarzian-like mechanism. The paper also discusses a holographic gravity interpretation, including a 3D gravity conjecture for , and suggests further exploration of higher-spin or -type symmetries and their bulk duals. Overall, the work establishes a unified framework for SYK-type systems with global symmetries, revealing new symmetry structures, spectral towers, and potential gravity duals that deepen the connection between quantum chaos and holography.

Abstract

In this paper, we study an SYK model and an SYK-like tensor model with global symmetry. First, we study the large expansion of the bi-local collective action for the SYK model with manifest global symmetry. We show that the global symmetry is enhanced to a local symmetry at strong coupling limit, and the corresponding symmetry algebra is the Kac-Moody algebra. The emergent local symmetry together with the emergent reparametrization is spontaneously and explicit broken. This leads to a low energy effective action. We evaluate four point functions, and obtain spectrum of our model. We derive the low energy effective action and analyze the chaotic behavior of the four point functions. We also consider the recent 3D gravity conjecture for our model. We also introduce an SYK-like tensor model with global symmetry. We first study chaotic behavior of four point functions in various channels for the rank-3 case, and generalize this into a rank- tensor model.

Paper Structure

This paper contains 38 sections, 305 equations, 9 figures.

Table of Contents

  1. Introduction
  2. SYK Models with Global Symmetry
  3. Bi-local Collective Actions
  4. Generalization to $q$ point interaction:
  5. Large $N$ Classical Solution
  6. Reparametrization and Kac-Moody Algebra
  7. Four point Functions
  8. Diagonalization of Quadratic Action in Large $N$
  9. Diagonalization of the Quadratic Action
  10. Kernel and Four Point Function
  11. Schwinger Dyson Equation for four point function:
  12. Diagrammatic Derivation of Two Point Function and Kernels
  13. Chaos
  14. Lypunov Exponent from Effective Action
  15. Contribution of Effective Action
  16. ...and 23 more sections

Figures (9)

  • Figure 1: Contributions to two point function self energy
  • Figure 2: Ladder diagrams. There are also diagrams with $(\alpha_3,\tau_3) \leftrightarrow (\alpha_4,\tau_4)$ with a relative minus sign.
  • Figure 3: The Vertices of the on-site and (non-local) hopping interactions.
  • Figure 4: Contribution to the Schwinger-Dyson equation for the two point function.
  • Figure 5: Dipoles $\bold{D}_{{\mathfrak{c}}}^{\alpha_1\alpha_2\alpha_3\alpha_4}$ of three colors
  • ...and 4 more figures