Quantum chaos dynamics in long-range power law interaction systems

Abstract

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