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29 March 2024
 
  » arxiv » 1704.2979

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Out-of-time-order Operators and the Butterfly Effect
Jordan S. Cotler ; Dawei Ding ; Geoffrey R. Penington ;
Date 10 Apr 2017
AbstractOut-of-time-order (OTO) operators have recently become popular diagnostics of quantum chaos in many-body systems. The usual way OTO operators are introduced is via a quantization of classical Lyapunov growth, which measures the divergence of classical trajectories in phase space due to the butterfly effect. However, it is not obvious how exactly the OTO captures the sensitivity of a quantum system to its initial conditions beyond the classical limit. In this paper, we analyze sensitivity to initial conditions in the quantum regime by recasting OTO operators for many-body systems using various formulations of quantum mechanics. Notably, we utilize the Wigner phase space formulation to derive an explicit $hbar$-expansion of the OTO to all orders for spatial degrees of freedom, and a large spin $1/s$-expansion for spin degrees of freedom. We find in each case that the leading term is exactly the Lyapunov growth for the classical limit of the system and argue that quantum corrections cause the OTO operator to saturate at around the scrambling time. These semiclassical expansions also provide a new way to numerically compute OTO operators perturbatively by solving a system’s classical equations of motion and systematically adding quantum corrections. We also express the OTO operator in terms of propagators and see from a different point of view how the OTO operator is a quantum generalization of the divergence of classical trajectories.
Source arXiv, 1704.2979
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