Quantum chaos dynamics in long-range power law interaction systems

Abstract

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 $α$. Three light cone structures of OTOC appear at $N = 1$: (1) logarithmic when $0.5<α\lesssim 0.8$, (2) sublinear power law when $0.8 \lesssim α\lesssim 1.5$ and (3) linear when $α\gtrsim 1.5$. The OTOC scales as $\exp(λt)/x^{2α} $ and $t^{2 α/ ζ} / x^{ 2 α} $ respectively beyond the light cones in the first two cases. When $α\geq 2$, 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 $ α\gtrsim 1.5$. This implies the locality is never fully recovered for finite $α$. Our result provides a unified physical picture for the chaos dynamics in long-range power law interaction system.