Quantum chaos dynamics in long-range power law interaction systems

We use out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: the number of qubits $N$ on each site (the onsite Hilbert space dimension) and the power law exponent $\alpha$. Three light cone structures of OTOC appear at $N = 1$: (1) logarithmic when $0.5<\alpha\lesssim 0.8$, (2) sublinear power law when $0.8 \lesssim \alpha \lesssim 1.5$ and (3) linear when $\alpha \gtrsim 1.5$. The OTOC scales as $\exp(\lambda t)/x^{2\alpha}$ and $t^{2 \alpha / \zeta} / x^{ 2 \alpha}$ respectively beyond the light cones in the first two cases. When $\alpha \geq 2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large $N$ limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $\alpha \gtrsim 1.5$. This implies the locality is never fully recovered for finite $\alpha$. Our result provides a unified physical picture for the chaos dynamics in long-range power law interaction system.