Non-Stabilizerness of Sachdev-Ye-Kitaev Model

TL;DR

The paper investigates non-stabilizerness (quantum magic) in the SYK model and its SYK2 variant by analyzing the Majorana spectrum and Stabilizer Rényi Entropy (SRE) of ground states and dynamical states. Using exact diagonalization and Monte Carlo sampling of Majorana strings, it finds a Gaussian Majorana spectrum for the chaotic SYK and a Laplace distribution for the integrable SYK2, with the SYK model exhibiting higher magic as evidenced by larger SRE that scales linearly with system size. Under unitary dynamics from stabilizer states, SYK2 equilibrates quickly while SYK maintains enhanced magic in the steady state, indicating qualitative distinctions in spectral statistics between chaotic and non-chaotic many-body systems. The work demonstrates that the Majorana spectrum and SRE provide robust signatures of chaos and non-stabilizerness, offering a resource-theoretic lens on quantum chaos.

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.

Figures (8)

• Figure 1: (a) The Majorana spectrum $\Pi(x)$ for the ground state of ${\rm SYK}_2$ model as a function of $x=\langle \Psi|\hat{\mu}(v)|\Psi\rangle$ is shown as a histogram in semi-logarithmic ($\log y$) scale. The spectrum clearly follows a Laplace distribution $\sim \exp(-|x|/b)$ . (b) The Majorana spectrum $\Pi(x)$ for the ground state of ${\rm SYK}$ model as a function of $x$ is shown as a histogram in semi-logarithmic ($\log y$) scale. The spectrum clearly follows a Gaussian distribution. Both (a, b) are computed from system size $N=12$ at half-filling.
• Figure 2: Stabilizer Rényi Entropy (SRE) for the ground state of the ${\rm SYK}_2$ and SYK models: (a) The SRE $M_{\alpha}$ and the filtered SRE $\tilde{M}_{\alpha}$ are shown as functions of the Rényi index $\alpha$ for the ${\rm SYK}_2$ and SYK models at system size $N = 12$. (b) The scaling of the SRE $M_2$ with system size $N$ is shown for both models and compared to the maximal entropy $N \log 2$. (c) The filtered SRE $\tilde{M}_{\alpha}$ for $\alpha = 2, 3, 4, 5$ is plotted as a function of $N$ for the ${\rm SYK}_2$ model. (d) The filtered SRE $\tilde{M}_{\alpha}$ for $\alpha = 2, 3, 4, 5$ is plotted as a function of $N$ for the SYK model. All results are averaged over disorder realizations, with standard deviations shown as error bars.
• Figure 3: (a-e) Time evolution of the Majorana spectrum $\Pi(x)$ after a quench at $t=0$ under ${\rm SYK}_2$ is shown. The long-time state follows a Laplace distribution ($\sim \exp(-|x|/b)$), as illustrated in (e). (f-j) Time evolution of the Majorana spectrum $\Pi(x)$ after a quench at $t=0$ under ${\rm SYK}$ is shown. The long-time state follows a Gaussian distribution, as illustrated in (j). The initial state is a product state given in Eq. \ref{['eq:initialstate']}. All the distributions are shown in semi-logarithmic ($\log y$) scale. This is shown for system size $N=10$ using ED method.
• Figure 4: (a) Time evolution of SRE $M_2[\Psi(t)]$ under ${\rm SYK}_2$ Hamiltonian is shown for different system sizes after quench at $t=0$. (b) Time evolution of $M_2[\Psi(t)]$ under ${\rm SYK}$ Hamiltonian is shown for different system sizes starting after quench at $t=0$. (c) Comparision of time evolution of SRE $M_2[\Psi(t)]$ under ${\rm SYK}_2$ and ${\rm SYK}$ Hamiltonian for $N=8$ system. (d) The SRE ($M_2[\Psi(t\gg 1)]$ of long-time states ($t\gg 1$) (saturation values) are plotted as a function of system sizes $N$ for both ${\rm SYK}_2$ and ${\rm SYK}$ model. The initial state is a product state as given in eqn.\ref{['eq:initialstate']}. The plot in (a-c) are shown in semi-logarithmic ($\log y$) scale and the plot in (d) are shown in linear scale.
• Figure 5: (a-e) The Majorana spectrum $\Pi(x)$ for five randomly chosen disorder realizations of the ${\rm SYK}_2$ model are shown here. (f-j) Similarly, $\Pi(x)$ for five randomly chosen disorder realizations of the ${\rm SYK}$ model are presented. The Gaussian (red) and exponential Laplace (blue) fittings are also shown in the histograms, indicating that each realisation of ${\rm SYK}$ model fits a Gaussian distribution, while the ${\rm SYK}_2$ model fits a Laplace distribution. The spectrum is computed from system size $N=8$.