Non-Stabilizerness of Sachdev-Ye-Kitaev Model

TL;DR

This paper investigates the quantum magic and non-stabilizerness of the SYK model, revealing that chaotic SYK exhibits higher magic and distinct spectral features compared to non-chaotic models, with implications for quantum chaos and information spreading.

Contribution

It provides a comparative analysis of the Majorana spectrum and stabilizer entropy in chaotic and non-chaotic SYK models, highlighting differences in magic and spectral distributions.

Findings

SYK ground state spectrum is Gaussian, indicating chaos.

SYK2 spectrum follows an exponential Laplace distribution.

SYK model exhibits higher stabilizer Renyi entropy, indicating more magic.

Abstract

We study the non-stabilizerness or quantum magic of the Sachdev-Ye-Kitaev ($\rm SYK$) model, a prototype example of maximally chaotic quantum matter. We show that the Majorana spectrum of its ground state, encoding the spreading of the state in the Majorana basis, displays a Gaussian distribution as expected for chaotic quantum many-body systems. We compare our results with the case of the $\rm SYK_2$ model, describing non-chaotic random free fermions, and show that the Majorana spectrum is qualitatively different in the two cases, featuring an exponential Laplace distribution for the $\rm SYK_2$ model rather than a Gaussian. From the spectrum we extract the Stabilizer Renyi Entropy (SRE) and show that for both models it displays a linear scaling with system size, with a prefactor that is larger for the SYK model, which has therefore higher magic. Finally, we discuss the spreading of quantun magic under unitary dynamics, as described by the evolution of the Majorana spectrum and the Stabilizer Renyi Entropy starting from a stabilizer state. We show that the SRE for the $\rm SYK_2$ model equilibrates rapidly, but that in the steady-state the interacting chaotic SYK model has more magic than the simple $\rm SYK_2$. Our results suggest that the Majorana spectrum is qualitatively distinct in chaotic and non-chaotic many-body systems.