Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry

TL;DR

This work analyzes scaling limits of the complex SYK model with $N$ fermions and $p$-body interactions, focusing on the large-$p$ limit and a double-scaling regime at fixed $\lambda=p^2/N$. The authors derive analytic expressions for the fermion two-point function $G(\tau)$ and the grand potential $\Omega$, revealing a Liouville-type equation governing symmetric fluctuations and a charge-induced antisymmetric component, and they show a precise connection to the conformal Schwarzian description in appropriate regimes. They demonstrate that the double-scaling results match the large-$p$ results in the $\lambda\to0$ limit, establishing consistency between the two analytical approaches, and they embed the SYK dynamics into a holographic dual described by a 2D Einstein-Maxwell-Dilaton theory with AdS$_2$ geometry, where the symmetric Green's function maps to the bulk metric and the antisymmetric part to the Maxwell field. The work further identifies the bulk dilaton profile and electric field, highlights the role of a $U(1)$ gauge field in the charged case, and provides a unified gravity–SYK framework for understanding holography beyond neutral Majorana setups, with implications for finite-temperature quantum gravity in low dimensions.

Abstract

We compare different limits of the Sachdev-Ye-Kitaev model of $N$ complex fermion with $p$-fermion interactions. First, we compute the fermion Green's function and free energy in the limit of large $N$ followed subsequently by the limit of large $p$. Next, we examine the `double-scaling' limit in which the large $N,p$ limits are taken at fixed $λ= p^2/N$. Earlier results on the latter limit are resummed for small $λ$, and shown to match our results for the first limit. We also describe the holographic match of our results to two-dimensional Jackiw-Teitelboim gravity with an additional $U(1)$ gauge field.

Figures (3)

• Figure 1: Blue lines represent exact values of $\ell_{\textrm{max}}$ and $v_{*}$ as functions of $Q_{0}$, obtained by numerically finding the maximum of the function $\ell(v)$ defined in Eq. (\ref{['hdef']}). The black dashed lines correspond to the theoretical asymptotics.
• Figure 2: Comparison of the exact complex SYK Green’s function (blue solid lines; obtained by numerically solving the Schwinger–Dyson equations (\ref{['SD']})) with the large $p$ approximation Eq. (\ref{['largepmu']}) (black dashed lines) at $\beta\mathcal{J}=10$ and $\beta\mu=0.2$. Panels correspond to $p=10$, $p=50$, and $p=100$. All curves are normalized to $G_{0}(\tau)$.
• Figure 3: Left: Numerical solution for the Green’s function for $p = 4$ and $\beta \mathcal{J} = 20$. The phase transition occurs between $\beta \mu = 9$ and $\beta \mu = 10$. Right: Numerical solution for the Green’s function for $p = 8$ and $\beta \mathcal{J} = 20$. The phase transition occurs between $\beta \mu = 3$ and $\beta \mu = 4$.