Chaos and random matrices in supersymmetric SYK

TL;DR

<3-5 sentence high-level summary>This paper investigates chaos and randomness in supersymmetric quantum systems by mapping the supersymmetric SYK model to the Wishart-Laguerre unitary ensemble (LUE) and computing spectral form factors, frame potentials, and correlation functions. The authors find that, unlike the non-supersymmetric case, the LUE exhibits no dip regime and a slower approach to Haar randomness, signaling a delayed onset of random-matrix behavior. Finite-temperature and higher-point form factors, together with frame-potential analyses, show that LUE is less efficient at scrambling and does not form a true k-design, consistent with 1-loop predictions from the super-Schwarzian theory. The results clarify how supersymmetry modifies late-time chaos and information spreading, with implications for holographic interpretations of SUSY chaotic systems.

Abstract

We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.

Figures (7)

• Figure 1: The $2$-point spectral form factor and its connected component for SYK with $N=24$ Majoranas at inverse temperature $\beta =1$, computed for 800 realizations of disorder. We observe the slope, dip, ramp, and plateau behaviors.
• Figure 2: The $2$-point spectral form factor and its connected piece for the supersymmetric SYK model with $N=24$ Majoranas at inverse temperature $\beta =1$, computed for 800 realizations of disorder. We observe the slope and plateau behaviors, while the ramp is obscured by the slow early-time decay of the 1-point function.
• Figure 3: On the left: the $2$-point spectral form factor and its connected component for the LUE at infinite temperature, as given in Eq. \ref{['eq:LUER2']}, plotted for different values of $L$ and normalized by the initial value $L^2$. We observe the slow $1/t$ decay down to the plateau value, hiding the linear ramp in the connected piece. On the right: the $2$-point spectral form factor for the GUE at infinite temperature, with a faster early-time decay exposing the ramp.
• Figure 4: The $2$-point spectral form factor for LUE at finite temperature, as given in Eq. \ref{['eq:R2beta']}, plotted for different values of $L$ and at different temperatures, normalized by the initial value. The plateau value depends on both $L$ and $\beta$, while the plateau time is just $L$ dependent.
• Figure 5: We show the first and second frame potentials for the LUE at infinite temperature at $L=1000$. The slow decay means we do not form a $k$-design at the dip time. For comparison, the Haar value is plotted in grey.