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arXiv:2602.07952

Higher-Order Corrections to Scrambling Dynamics in Brownian Spin SYK Models

Tingfei Li, Miao Wang, Jianghui Yu

quant-phcond-mat.stat-mechhep-thSubmitted Feb 8, 2026• Announced Feb 10, 2026• 0 citations• Revised Feb 19, 2026 (v2)
v1 · Feb 8, 2026v2 · Feb 19, 2026v3 · Jul 13, 2026
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Abstract

We investigate operator growth in a Brownian spin Sachdev--Ye--Kitaev (SYK) model with random all-to-all interactions, focusing on the full operator-size distribution. For Hamiltonians containing interactions of order two up to $L$, we derive a closed master equation for the Pauli-string expansion coefficients and recast their dynamics into a generating-function formulation suitable for the large-$N$ limit. This approach allows us to diagonalize the leading-order evolution operator explicitly and obtain exact solutions for arbitrary initial operator distributions, including the effects of decoherence. Going beyond leading order, we develop a systematic $1/N$ expansion that captures higher-order corrections to the operator-size dynamics and the late-time behavior. Our results demonstrate that higher-order effects play a crucial role in operator scrambling and that the full operator-size distribution provides a more refined probe of quantum chaos in Brownian and open quantum systems.

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Abstract We study operator dynamics in Brownian quantum many-body models with q -local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev–Ye–Kitaev model and the spin model for general q . This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large N limit for arbitrary initial conditions and q . The distributions for both models take the same form, related to the χ -squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below q − 2 for the Majorana model or q − 1 for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all q decays exponentially as t e − t , much slower than all other operators, which decay as e − t . At finite N , the size distribution exhibits modulo-dependent branching within a symmetry sector for the q ⩾ 8 Majorana model and the q ⩾ 4 spin model. Our results reveal universal features of operator dynamics in q -local quantum many-body systems.
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Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
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