On large $q$ expansion in the Sachdev-Ye-Kitaev model

TL;DR

This work advances the analytic control of the SYK$_q$ model by computing the next $1/q^{2}$ correction to the thermal two-point function and the corresponding free-energy correction. By solving coupled equations for the $1/q$ and $1/q^{2}$ components of the Green function, it provides explicit forms for $g(\tau)$ and $h(\tau)$ and a closed expression for the $1/q^{2}$ correction to $G(\tau)$, improving the UV/IR interpolation. The results show that an exponentiated large-$q$ ansatz yields excellent agreement with numerical Schwinger–Dyson solutions and enable a detailed free-energy and Schwarzian analysis, including a Padé approximant for the Schwarzian coefficient $\alpha_S(q)$. These findings sharpen the analytic handle on SYK$_q$ dynamics and open paths to applications in related SYK-type and tensor models, as well as higher-dimensional generalizations.

Abstract

We consider the Sachdev-Ye-Kitaev (SYK) model where interaction involves $q$ fermions at a time. We find the next order correction to the thermal two-point function in the large $q$ expansion. Using this result we find the next order correction to the SYK free energy.

Figures (2)

• Figure 1: (Color online) Plots of the numerical solution and the large $q$ approximations for $G(\theta)$, $\theta=2\pi \tau/\beta$ at $\beta J = 20,50,100,1000$ and $q=4$. The black solid line is the numerical solution for the Schwinger-Dyson equation (\ref{['SD']}). The blue dash-dotted line is the large $q$ approximation (\ref{['Gtau']}) with $1/q^2$ term. The blue dashed line is the exponentiated two-point function (\ref{['Gimpr']}).
• Figure 2: (Color online) Plot of $\alpha_{S}$ as a function of $q$. The black circles correspond to numerical results adapted from Maldacena:2016hyu. The blue solid line corresponds to the two-sided Pade approximation (\ref{['padeas']}).