Non-stabilizerness of Sachdev-Ye-Kitaev model

We study the non-stabilizerness or quantum magic of the Sachdev-Ye-Kitaev ( \mathrm{SYK} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>Y</mml:mi> <mml:mi>K</mml:mi> </mml:mstyle> </mml:math> ) 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 \mathrm{SYK_2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>Y</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:math> 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 \mathrm{SYK_2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>Y</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:math> 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 \mathrm{SYK_2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>Y</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:math> model equilibrates rapidly, but that in the steady-state the interacting chaotic SYK model has more magic than the simple \mathrm{SYK_2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>Y</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:math> . Our results suggest that the Majorana spectrum is qualitatively distinct in chaotic and non-chaotic many-body systems.