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

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 (SYK) 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 $\chi$-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 $q\ge 8$ Majorana model and the $q\ge 4$ spin model. Our results reveal universal features of operator dynamics in $q$-local quantum many-body systems.

31 pages, 7 figures

References in corpus (7)

• Thermalization and its mechanism for generic isolated quantum systems
• Random Quantum Circuits
• Scrambling Dynamics and Out-of-Time Ordered Correlators in Quantum Many-Body Systems: a Tutorial
• Dynamical Transition of Operator Size Growth in Quantum Systems Embedded in an Environment
• Note on entropy dynamics in the Brownian SYK model
• Operator Size Distribution in Large $N$ Quantum Mechanics of Majorana Fermions
• Complexity is not Enough for Randomness

Cited by in corpus (2)

• Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos
• Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians