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Complex Sachdev-Ye-Kitaev model in the double scaling limit

Micha Berkooz, Vladimir Narovlansky, Himanshu Raj

TL;DR

The paper exactly solves the complex SYK model in the double scaling limit with a $U(1)$ symmetry, providing a complete spectrum and exact 2-point and 4-point functions across all energy scales via chord diagrams. It develops a comprehensive diagrammatic framework, including canonical and grand partition functions with chemical potential, and extends to fixed-charge ensembles and a $U(M)$ generalization. It also analyzes heavy operator insertions that effectively partition spacetime into interfaces, while revealing how light states may traverse these interfaces. The results connect to spectral asymmetry, Schwarzian gravity limits, and holographic interpretations, offering precise control over charge and operator content in a highly chaotic quantum system. Overall, the work delivers a universal, exact treatment of observables in the double-scaled complex SYK model with broad implications for chaotic quantum systems and their gravitational duals.

Abstract

We solve for the exact energy spectrum, 2-point and 4-point functions of the complex SYK model, in the double scaling limit at all energy scales. This model has a $U(1)$ global symmetry. The analysis shows how to incorporate a chemical potential in the chord diagram picture, and we present results for the various observables also at a given fixed charge sector. In addition to matching to the spectral asymmetry, we consider an analogous asymmetry measure of the 2-point function obeying a non-trivial dependence on the operator's dimension. We also provide the chord diagram structure for an SYK-like model that has a $U(M)$ global symmetry at any disorder realization. We then show how to exactly compute the effect of inserting very heavy operators, with formally infinite conformal dimension. The latter separate the gravitational spacetime into several parts connected by an interface, whose properties are exactly computable at all scales. In particular, light enough states can still go between the spaces. This behavior has a simple description in the chord diagram picture.

Complex Sachdev-Ye-Kitaev model in the double scaling limit

TL;DR

The paper exactly solves the complex SYK model in the double scaling limit with a symmetry, providing a complete spectrum and exact 2-point and 4-point functions across all energy scales via chord diagrams. It develops a comprehensive diagrammatic framework, including canonical and grand partition functions with chemical potential, and extends to fixed-charge ensembles and a generalization. It also analyzes heavy operator insertions that effectively partition spacetime into interfaces, while revealing how light states may traverse these interfaces. The results connect to spectral asymmetry, Schwarzian gravity limits, and holographic interpretations, offering precise control over charge and operator content in a highly chaotic quantum system. Overall, the work delivers a universal, exact treatment of observables in the double-scaled complex SYK model with broad implications for chaotic quantum systems and their gravitational duals.

Abstract

We solve for the exact energy spectrum, 2-point and 4-point functions of the complex SYK model, in the double scaling limit at all energy scales. This model has a global symmetry. The analysis shows how to incorporate a chemical potential in the chord diagram picture, and we present results for the various observables also at a given fixed charge sector. In addition to matching to the spectral asymmetry, we consider an analogous asymmetry measure of the 2-point function obeying a non-trivial dependence on the operator's dimension. We also provide the chord diagram structure for an SYK-like model that has a global symmetry at any disorder realization. We then show how to exactly compute the effect of inserting very heavy operators, with formally infinite conformal dimension. The latter separate the gravitational spacetime into several parts connected by an interface, whose properties are exactly computable at all scales. In particular, light enough states can still go between the spaces. This behavior has a simple description in the chord diagram picture.

Paper Structure

This paper contains 20 sections, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Outline and summary of new results
  3. Review of the complex Sachdev-Ye-Kitaev model
  4. Definition of the model
  5. Summary of known results
  6. Thermodynamics
  7. Two-point function
  8. Computation using chord diagrams
  9. Chord diagrams and the real SYK model
  10. Chord diagrams and the complex SYK model
  11. Large $N$ simplification
  12. Partition function
  13. Canonical partition function
  14. Grand canonical partition function
  15. Partition function at different scalings of the chemical potential
  16. ...and 5 more sections

Figures (6)

  • Figure 1: An example of a typical chord diagram.
  • Figure 2: Chord diagram representation of the generic trace \ref{['general_trace']}. The chords are oriented to go from a $\psi$ insertion to a $\bar{\psi}$ insertion having the same index set $I$.
  • Figure 3: Determining the sign of a diagram. Only in this figure the chords are shown in different colors. The chords to which $j$ belongs are shown in orange, while the ones to which $k$ belongs are in blue. In fig. \ref{['fig:signs_more_general']} we draw specific orientations for concreteness, but the argument in the text is independent of the orientations of the chords.
  • Figure 4: Rules for evaluating oriented chord diagrams (when an orientation is not shown, it means that it does not matter). The values when we have a chemical potential, to be used later (which are derived in section \ref{['sec:partition_function']}), are also shown.
  • Figure 5: An oriented chord diagram contributing to $\langle{\rm tr} H^{8}\rangle_{J}$ and the corresponding unoriented chord diagram.
  • ...and 1 more figures