Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correlators of double scaled SYK at one-loop

Kazumi Okuyama, Kenta Suzuki

TL;DR

The paper computes one-loop corrections in the double-scaled SYK model using both chord diagrams and a Liouville-type action, deriving explicit one-loop corrections to the two-point and uncrossed four-point functions and reproducing the exact OTOC at arbitrary temperatures within this approximation. It unifies the saddle-point analysis with Liouville theory by linking the one-loop data to fluctuations around the Liouville classical solution, and shows that the off-diagonal energy-exchange term I_{12} encodes bulk energy fluctuations. In the low-temperature limit, the results consistently match Schwarzian theory, and the OTOC analysis yields the correct Lyapunov behavior. The work clarifies the role of Liouville dynamics in DSSYK and outlines directions toward a deeper bulk dual interpretation and finite λ generalizations.

Abstract

In this paper, we study one-loop contributions in the double-scaling limit of the SYK model from the chord diagrams and Liouville type effective action. We compute and clarify the meaning of each component consisting of the one-loop corrections for the two- and time-ordered four-point functions of light operators. We also reproduce the exact expression of the out-of-time-ordered four-point function at arbitrary temperatures within the one-loop level, which were previously computed from different methods.

Correlators of double scaled SYK at one-loop

TL;DR

The paper computes one-loop corrections in the double-scaled SYK model using both chord diagrams and a Liouville-type action, deriving explicit one-loop corrections to the two-point and uncrossed four-point functions and reproducing the exact OTOC at arbitrary temperatures within this approximation. It unifies the saddle-point analysis with Liouville theory by linking the one-loop data to fluctuations around the Liouville classical solution, and shows that the off-diagonal energy-exchange term I_{12} encodes bulk energy fluctuations. In the low-temperature limit, the results consistently match Schwarzian theory, and the OTOC analysis yields the correct Lyapunov behavior. The work clarifies the role of Liouville dynamics in DSSYK and outlines directions toward a deeper bulk dual interpretation and finite λ generalizations.

Abstract

In this paper, we study one-loop contributions in the double-scaling limit of the SYK model from the chord diagrams and Liouville type effective action. We compute and clarify the meaning of each component consisting of the one-loop corrections for the two- and time-ordered four-point functions of light operators. We also reproduce the exact expression of the out-of-time-ordered four-point function at arbitrary temperatures within the one-loop level, which were previously computed from different methods.
Paper Structure (17 sections, 161 equations, 1 figure)

This paper contains 17 sections, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of double scaled SYK
  3. Saddle point computation of correlators
  4. Partition function
  5. Two-point function
  6. Uncrossed four-point function
  7. Relation to the energy fluctuation
  8. One-loop correction from Liouville theory
  9. One-loop correction of two-point function
  10. Uncrossed four-point function
  11. Out-of-time-order correlators
  12. Low temperature limit
  13. One-loop correction from Schawrzian mode
  14. Conclusions and outlook
  15. Alternative derivation of one-loop determinant
  16. ...and 2 more sections

Figures (1)

  • Figure 1: An example of a chord diagram for $k=8$. The black circle represents a trace over the fermions. Each orange dot represents an insertion of a Hamiltonian and each blue line denotes a contraction of capital index $I$.