Quantum chaos dynamics in long-range power law interaction systems

TL;DR

This paper investigates how chaos propagates in one-dimensional long-range systems using OTOC, revealing different light cone structures depending on parameters and showing a transition from nonlocal to local behavior.

Contribution

It maps OTOC evolution to a classical stochastic process and derives exact master equations, providing a unified understanding of chaos dynamics in long-range power law interacting systems.

Findings

Identifies three light cone regimes based on the power law exponent .

Derives explicit scaling forms of OTOC beyond light cones.

Shows the transition to short-range-like diffusive behavior at  .

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 $\alpha$. 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.