Stabilizer Rényi Entropy and its Transition in the Coupled Sachdev-Ye-Kitaev Model

Abstract

Quantum entanglement and quantum magic are two distinct fundamental resources that enable quantum systems to exhibit complex phenomena beyond the capabilities of classical computer simulations. While quantum entanglement has been extensively used to characterize both equilibrium and dynamical phases, the study of quantum magic, typically quantified by the stabilizer Rényi entropy (SRE), remains largely limited to numerical simulations of moderate system sizes. In this Letter, we establish a general framework for analyzing the SRE in solvable Sachdev-Ye-Kitaev (SYK) models in the large-$N$ limit, which enables the application of the saddle-point approximation. Applying this method to the Maldacena-Qi coupled SYK model, we identify a series of first-order transitions of the SRE as the temperature is tuned. In particular, we uncover an intrinsic transition of the SRE that cannot be detected through thermodynamic quantities. We also discuss the theoretical understanding of the SRE in both the high-temperature and low-temperature limits. Our results pave the way for studying the SRE in strongly correlated fermionic systems in the thermodynamic limit and suggest a new class of transitions for which the SRE serves as an order parameter.

Figures (4)

• Figure 1: A schematic of our main results. The Maldacena-Qi coupled SYK model consists of two identical copies of the SYK model with the same random couplings $J_{ijkl}$ and a direct hopping $\mu$. In thermal equilibrium, the model exhibits a first-order Hawking-Page transition at $\beta^*_{\text{HP}}$ for $\mu\ll J$. We find that the (second) stabilizer Rényi entropy $\tilde{M}_2(\beta)$ exhibits its own intrinsic first-order transition at $\beta^*_{\text{SRE}}$, which is not reflected in thermodynamic quantities. Solid lines correspond to the dominant saddle, while dashed lines denote unstable saddles.
• Figure 2: An illustration of the path-integral representation of the stabilizer Rényi entropy, either with extensive operator insertions (Eq.\ref{['eq:fourop']}) or with fluctuating phases (Eq.\ref{['eq:final']}). Lines of different colors correspond to different replicas of the density matrix.
• Figure 3: (a) Numerical results for $M_2(\beta)$, obtained by iteratively solving the saddle-point equations \ref{['eq:saddle']} for $\mu/J \in {0,0.1,0.2}$. The orange triangles and purple diamonds correspond to solutions with different initializations, favoring the high- and low-temperature solutions, respectively. (b) The Green's function $G^{(11)}_{LL}(\tau,\tau')$ is plotted at $\beta J=17$ and $\beta J=21$ for $\mu/J=0.1$, clearly exhibiting qualitatively different features near $\tau=0^+$ and $\tau'=\beta^-$.
• Figure 4: Numerical results for (a) $S_2(\beta)$, obtained by solving the saddle-point equations for the partition function Maldacena:2018lmt, and (b) $\tilde{M}_2(\beta)$, computed using $M_2(\beta)$ from FIG. \ref{['fig:numerics']}. The results reveal three first-order transitions of the SRE for $\mu/J=0.1$, occurring at $\beta^*_{\text{HP}}/2$, $\beta^*_{\text{SRE}}$, and $\beta^*_{\text{HP}}$.