Stabilizer Entropy and entanglement complexity in the Sachdev-Ye-Kitaev model

TL;DR

The paper investigates how entanglement and non-stabilizerness, quantified by Stabilizer Rényi Entropy (SRE), co-evolve in the interpolated SYK-4+SYK-2 model across ground and middle-energy states. By employing entanglement entropy, entanglement spectrum statistics, and anti-flatness measures, it reveals Haar-like universal features in the SYK-4 sector that are fragile in the ground state under any finite SYK-2 perturbation but robust in high-energy states for g<1, while SYK-2 remains non-universal. A KL-fidelity based probe uncovers a complexity transition with critical interpolation strength g_c that scales as g_c ~ N^{-3/4} in the ground state, indicating instability of the SYK-4 phase to quadratic perturbations in the thermodynamic limit. The findings highlight a nuanced landscape where high-energy states display more universal behavior than ground states, and where SRE captures symmetry-class structure (8-fold) and non-local magic that is not fully captured by entanglement alone. This has implications for understanding quantum chaos, holography, and quantum information processing in SYK-like systems.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is of paramount importance for the understanding of both strange metals and a microscopic theory of two-dimensional gravity. We study the interplay between Stabilizer Rényi Entropy (SRE) and entanglement entropy in both the ground state and highly excited states of the SYK4+SYK2 model interpolating the highly chaotic four-body interactions model with the integrable two-body interactions one. The interplay between these quantities is assessed also through universal statistics of the entanglement spectrum and its anti-flatness. We find that SYK4 is indeed characterized by a complex pattern of both entanglement and non-stabilizer resources while SYK2 is non-universal and not complex. We discuss the fragility and robustness of these features depending on the interpolation parameter.

Figures (15)

• Figure 1: Schematic representation of the interpolated SYK-4 + SYK-2 model defined in Eq. (5). At $g=0$, the model reduces to SYK-4, a maximally chaotic system characterized by universal entanglement properties. At $g=1$, the model reaches the SYK-2 limit which behaves like a disorder free-fermion system. In the ground state, SYK-4 features are fragile under finite values of the interpolation parameter $g$, whereas those of SYK-2 remain robust. By contrast, SYK-4 behavior persists at high energies for all $g \ne 1$, while making SYK-2 regime fragile.
• Figure 2: Upper panels: Averaged bipartite von Neumann entanglement entropy, defined in Eq. \ref{['eq-VN']}, for the SYK-4 + SYK-2 model as a function of the interpolation parameter $g$. Shaded regions indicate the standard deviation over $M$ disorder realizations. Lower panels: Finite-size scaling of the relative entanglement entropy gap, defined in Eq. \ref{['eq-relativegap']}, along with its linear extrapolation to the thermodynamic limit. This figure highlights the fundamental difference in entanglement (i.e., non-local correlations) between the ground state (GS) and a middle-spectrum eigenstate (MS) across the interpolation. The left column shows results for the GS, while the right column corresponds to a MS.
• Figure 3: Normalized reduced density matrix (RDM) eigenvalues of $f = 1/2$ subsystem-to-system. These are average values over many iterations \ref{['SMstatistics']} across a single bipartition. One obtains similar results choosing other bipartitions. The reference value is the Marchenko-Pastur (M-P) distribution $\eta^{\rm Haar}(x)$ in Eq. \ref{['etaHaar']}. We define $\eta_{k} = k /d$ and $x_{k} = (1/2) \sqrt{ \lambda_{k} d}$ where $d = 2^{N/4}$ for $f=1/2$, while the binning step for the interpolation parameter has been set to $\epsilon = 0.01$. The left column shows results for the ground state (GS), while the right column corresponds to a middle-of-spectrum (MS) eigenstate.
• Figure 4: Left panel: KL divergence–based fidelity as a function of the interpolation parameter $g$ for different binning choices $\epsilon$, with fixed system size $N = 20$. The rescaled fidelity diagnostic exhibits robust behavior across $\epsilon$, justifying the use of $\epsilon = 0.01$ to extract the transition point. The solid black line represents a degree-10 polynomial fit used to locate the peak (marked by the dashed vertical line). Right panel: Finite-size scaling of the extracted transition point $g_c$, obtained from the location of the fidelity maximum. The fit reveals a power-law scaling, indicating that the SYK-4 ground state becomes unstable under arbitrarily small quadratic perturbations in the thermodynamic limit. In between $N^{-1}$ and $N^{-1/2}$ is where Ref. PhysRevLett.125.196602 identified the potential transition to be located.
• Figure 5: ESS of the GS RDM eigenvalues of the $H_g$ model for different values of $g$. We superimpose the analytical curves for the Wigner-Dyson (dashed) and Poisson (blue continuous) distribution for comparison. System size $N = 22$ and number of realizations is $M = 100$. The number of bins used for the histogram is 100. The colored numbers represent the KL divergence of data against known distributions described in Appendix \ref{['app:HamS']}.