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.
Physical Review B