Operator Dynamics in Brownian Quantum Circuit

We explore the operator dynamics in a random $N$-spin model with pairwise interactions (Brownian quanum circuit). We introduce the height $h$ of an operator to characterize its spatial extent, and derive the master equation of the height probability distribution. The study of an initial simple operator with $h = 1$ (minimal nonzero height) shows that the mean height, which is proportional to the squared commutator, has an initial exponential growth. It then slows down around the scrambling time $\sim\log N$ and finally saturates to a steady state in a manner similar to the logistic function. The deviation to the logistic function is due to the large fluctuations (order $N$) in the intermediate time. Moreover, we find that the exponential growth rate (quantum Lyapunov exponent) is smaller for initial operator with $\langle h\rangle\gg 1$. Based on this observation, we propose that the chaos bound at finite temperature can be produced by an initial operator whose height distribution is biased towards higher operators. We numerically test the power law initial height distribution $1/h^{\alpha} $ in a Brownian circuit with number of spin $N=10000$ and show that the Lyapunov exponent is linearly constrained by $\alpha$ before reaching the infinite temperature value.