SYK Model, Chaos and Conserved Charge

TL;DR

This work studies chaos in the SYK model with complex fermions at finite chemical potential $\mu$ in the large-$q$ limit. By solving the Schwinger-Dyson equations and computing the retarded kernel, it shows that the Lyapunov exponent $\lambda_L$ is a sensitive function of $\mu$, with suppression occurring when $\mu$ is small compared to the disorder scale $J$ and the UV data $(\beta J,\beta \mu)$ renormalizing to an IR coupling $\beta\tilde{J}$. The analysis yields an explicit_IR relation $\beta\tilde{J}=\pi\nu/\cos(\pi\nu/2)$ and $\lambda_L=\frac{2\pi}{\beta}\nu$, revealing how $\mu$ tunes chaos from maximal to vanishing chaos; these results extend to flavoured complex fermions, where a large flavor number further suppresses chaos via an effective coupling $J_{ ext{eff}}$. The findings offer a controlled mechanism to dial chaos in solvable SYK-type models and have potential implications for holography and bulk gauge-field interpretations of chaos suppression.

Abstract

We study the SYK model with complex fermions, in the presence of an all-to-all $q$-body interaction, with a non-vanishing chemical potential. We find that, in the large $q$ limit, this model can be solved exactly and the corresponding Lyapunov exponent can be obtained semi-analytically. The resulting Lyapunov exponent is a sensitive function of the chemical potential $μ$. Even when the coupling $J$, which corresponds to the disorder averaged values of the all to all fermion interaction, is large, values of $μ$ which are exponentially small compared to $J$ lead to suppression of the Lyapunov exponent.

Figures (4)

• Figure 1: A diagrammatic representation of $\Sigma$. Each vertex is worth of strength $J$, and $\left( \frac{q}{2}-1 \right)$ propagators run inside the loop in each direction. The direction of the arrows correlate with the sign of $\tau$ in the argument of the propagators. The overall direction of the diagram, from left to right, selects out two additional propagators running in this direction and hence the corresponding powers of $G$.
• Figure 2: A diagrammatic representation of the four point function calculation, in the large $N$ limit. First, only the ladder diagrams contribute, as shown in the first row here. Second, from the structure of the diagrams, one obtains an iterative process to generate ${\cal F}_{n+1}$ from ${\cal F}_n$, composing with a kernel.
• Figure 3: The Lyapunov exponent $\lambda$ is normalised and takes values between 0 and 1. This figure shows dependence of $\lambda$ on $\beta J$ for different values of $\beta\mu$
• Figure 4: The Lyapunov exponent $\lambda$ is again normalised and takes values between 0 and 1. This figure shows dependence of $\lambda$ on $\beta\mu$ for different values of $\beta J$