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Non-Stabilizerness of Sachdev-Ye-Kitaev Model
by Surajit Bera, Marco Schirò
Submission summary
| Authors (as registered SciPost users): | Surajit Bera |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2502.01582v3 (pdf) |
| Date accepted: | Dec. 2, 2025 |
| Date submitted: | Nov. 13, 2025, 9:31 p.m. |
| Submitted by: | Surajit Bera |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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.
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Author comments upon resubmission
List of changes
(1) Based on referee 2's comments 1 and 2, we have added the following text on page 10 :
''We note that for random gaussian fermions an efficient algorithm has been proposed to compute
the SRE [41], which could be in principle adapted to compute the Majorana spectrum.
Here, we limit our analysis to small system sizes to perform a fair comparison with the
SYK4 model, which is the main focus of our work. Even with these small sizes, the
agreement with the Laplace distribution for the Majorana spectrum of the SYK2 model is
excellent. Our results suggests therefore that the Laplace distribution is a generic feature
of the Majorana spectrum for random Gaussian states. We note that obtaining this result
analytically is far from trivial, since even if the state is Gaussian and one can use the Wick
theorem to decompose the string of fermions, sampling is still required over a large number
of contributions. Whether analytical progress is possible in this direction is an exciting
question which is therefore left for future studies.''
(2) According to referee 2's suggestions (comment 2) and referee 3's requested change, we elaborate briefly on the overlap with ref 73 and ref 74 in the "Note added" section. The added text reads as below:
''Upon completion of this work, we became aware of Refs. [73,74], which also have partial overlap with our study of the non-stabilizerness of the Sachdev–Ye–Kitaev model. We briefly discuss the extent of this overlap below. Ref. [73] investigates the unitary dynamics of the stabilizer Rényi entropy (SRE) for the SYK model in the spin representation with total Sz = 0, which directly corresponds to our complex SYK model at half-filling and shows very similar steady-state values. Ref. [74] examines the ground- state SRE of the Majorana SYK model, which can be compared to our results for the complex SYK model; their results indicate a slightly lower SRE for the Majorana case, though both are of comparable magnitude''.
(3) According to referee 2's suggestion (comment 4), we have now incorporated the correct reference for SYK and related literature on page 2-3.
(4) According to referee 3's requested change, we have corrected the sentence on page 4, which reads now : 'On the other hand, odd-parity strings can act as logical operations' .
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