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arXiv:2408.11737 · Journal of Physics A Mathematical and Theoretical

Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models

Shenglong Xu

Published Jan 8, 2025• 5 citations
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Abstract

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.

DOI
10.1088/1751-8121/ada7ac

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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.
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 LLL, 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-NNN 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/N1/N1/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.
A bstract We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q -body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q . With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q , leading to classical dynamics of the operator size growth with dissipation.
We show that operator dynamics in U(1) symmetric systems are constrained by two independent conserved charges and construct a non-Abelian operator size basis that respects both, enabling a symmetry-resolved characterization of operator growth. The non-Abelian operator size depends on the operator's nonlocal structure and is organized by an SU(2) algebra. Operators associated with large total angular momentum are relatively simple, while those with small angular momentum are more complex. Operator growth is thus characterized by a reduction in angular momentum and can be probed using out-of-time-ordered correlators. Using the charge-conserving Brownian Sachdev-Ye-Kitaev model, we derive an exact classical master equation that governs the size distribution, the distribution of an operator expanded in this basis, for arbitrary system sizes. The resulting dynamics reveal that the size distribution follows a chi-squared form, with the two conserved charges jointly determining the overall time scale and the shape of the distribution. In particular, single-particle operators retain a divergent peak at large angular momentum throughout the time evolution.
Abstract In this work, we introduce a symmetry-based approach to study the scrambling and operator dynamics of Brownian SYK models at large finite N and in the infinite N limit. We compute the out-of-time-ordered correlator (OTOC) in the Majorana model without charge conservation and the complex model with charge conservation, and demonstrate that in both models taking the random average of the couplings gives rise to emergent symmetry structures. The random averaging exactly maps the operator dynamics of the Majorana model and the complex model to the imaginary time dynamics of an SU(2) spin and an SU(4) spin respectively, which become solvable in the large N limit. Furthermore, the symmetry structure drastically reduces the size of the Hilbert space required to calculate the OTOC from exponential to linear in N, providing full access to the operator dynamics at all times for large finite N. In the case of the complex model with charge conservation, using this approach, we obtain the OTOC within each charge sector both numerically at finite N and analytically in the large N limit. We find that the time scale of the scrambling dynamics for all times and in each sector is characterized by the charge density. Furthermore, after proper rescaling, the OTOC corresponding to different finite charge densities collapses into a single curve at large finite N. In the large N limit, the rescaled OTOCs at finite density are described by the same hydrodynamic equation as in the Majorana case.
A bstract Under the Heisenberg evolution in chaotic quantum systems, initially simple operators evolve into complicated ones and ultimately cover the whole operator space. We study the growth of the operator “size” in this process, which is related to the out-of-time-order correlator (OTOC). We derive the full time evolution of the size distribution in large N quantum mechanics of Majorana fermions. As examples, we apply the formalism to the Brownian SYK model (infinite temperature) and the large q SYK model (finite temperature).
A bstract We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of [1]. Our results are based on the study of the dissipative q -body Sachdev-Ye-Kitaev (SYK q ) model, governed by the Markovian dynamics. We introduce a notion of “operator size concentration” which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ( a n and b n ) in the large q limit. Our results corroborate with the semi-analytics in finite q in the large N limit, and the numerical Arnoldi iteration in finite q and finite N limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.