Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Spectrum in the Sachdev-Ye-Kitaev Model

Joseph Polchinski, Vladimir Rosenhaus

TL;DR

This work analyzes the SYK model by solving the Schwinger-Dyson equations for the two-point function and constructing the four-point function from the resulting kernel. It reveals a spectrum of two-particle states with both continuous and discrete components, enabling a spectral decomposition of the four-point function. The authors exploit SL(2,R) symmetry to generate a complete eigenbasis, derive explicit forms for the eigenfunctions, and compute the discrete contribution to the four-point function in closed form, while the continuous part remains to be fully integrated. The results illuminate IR conformal structure, UV-induced conformal symmetry breaking, and connections to holography via AdS2/CFT1 dynamics, including the emergence of chaos with maximal Lyapunov exponent in the SYK model.

Abstract

The SYK model consists of $N\gg 1$ fermions in $0+1$ dimensions with a random, all-to-all quartic interaction. Recently, Kitaev has found that the SYK model is maximally chaotic and has proposed it as a model of holography. We solve the Schwinger-Dyson equation and compute the spectrum of two-particle states in SYK, finding both a continuous and discrete tower. The four-point function is expressed as a sum over the spectrum. The sum over the discrete tower is evaluated.

The Spectrum in the Sachdev-Ye-Kitaev Model

TL;DR

This work analyzes the SYK model by solving the Schwinger-Dyson equations for the two-point function and constructing the four-point function from the resulting kernel. It reveals a spectrum of two-particle states with both continuous and discrete components, enabling a spectral decomposition of the four-point function. The authors exploit SL(2,R) symmetry to generate a complete eigenbasis, derive explicit forms for the eigenfunctions, and compute the discrete contribution to the four-point function in closed form, while the continuous part remains to be fully integrated. The results illuminate IR conformal structure, UV-induced conformal symmetry breaking, and connections to holography via AdS2/CFT1 dynamics, including the emergence of chaos with maximal Lyapunov exponent in the SYK model.

Abstract

The SYK model consists of fermions in dimensions with a random, all-to-all quartic interaction. Recently, Kitaev has found that the SYK model is maximally chaotic and has proposed it as a model of holography. We solve the Schwinger-Dyson equation and compute the spectrum of two-particle states in SYK, finding both a continuous and discrete tower. The four-point function is expressed as a sum over the spectrum. The sum over the discrete tower is evaluated.

Paper Structure

This paper contains 12 sections, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two-point function
  3. Spectrum
  4. Eigenvalues
  5. $SL(2,\mathcal{R})$ and all eigenvectors
  6. Directly finding the eigenvectors
  7. A complete set of eigenvectors
  8. Four-point function
  9. Integrals in eigenvalue computation
  10. Eigenvectors and boundary terms
  11. Integral in eigenvector computation
  12. Integrals of products of Bessel functions

Figures (2)

  • Figure 1: The line with a box is the full two-point function, while the solid line is the free two-point function. (a) The self-energy $\Sigma(t_1, t_2)$ in terms of the two-point function $G(t_1, t_2)$. (b) Some of the Feynman diagrams making up the two-point function. (c) The Schwinger-Dyson equation for the two-point function. Iterating generates the sum in (b).
  • Figure 2: (a) The four-point function is given by a sum of ladder diagrams, such as the one shown. (b) These ladder diagrams are generated by iterating the Schwinger-Dyson equation (note: the propagators are really the dressed propagators; we have suppressed the box on the line that it is meant to indicate this).