Out-of-time-order Operators and the Butterfly Effect
arXiv:1704.02979 · doi:10.1016/j.aop.2018.07.020
Abstract
Out-of-time-order (OTO) operators have recently become popular diagnostics of quantum chaos in many-body systems. The usual way they 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 they capture 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 -expansion of the OTO operator for spatial degrees of freedom, and a large spin -expansion for spin degrees of freedom. We find in each case that the leading term is the Lyapunov growth for the classical limit of the system and argue that quantum corrections become dominant at around the scrambling time, which is also when we expect the OTO operator to saturate. We also express the OTO operator in terms of propagators and see from a different point of view how it is a quantum generalization of the divergence of classical trajectories.
9+2 pages, 1 figure. Error corrected regarding phase space functions satisfying classical equations of motion
References in corpus (3)
Cited by in corpus (55)
- Does scrambling equal chaos?
- Scrambling and entanglement spreading in long-range spin chains
- Krylov complexity in saddle-dominated scrambling
- Chaos signatures in the short and long time behavior of the out-of-time ordered correlator
- Bridging entanglement dynamics and chaos in semiclassical systems
- Gauging classical and quantum integrability through out-of-time ordered correlators
- Treelike interactions and fast scrambling with cold atoms
- Semiclassical theory of out-of-time-order correlators for low-dimensional classically chaotic systems
- Scrambling and Complexity in Phase Space
- Universal level statistics of the out-of-time-ordered operator
- Quantum Chaos and the Correspondence Principle
- Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest time-scale
- Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices
- Bounds in Nonequilibrium Quantum Dynamics
- Pedagogical introduction to SYK model and 2D Dilaton Gravity
- Spread complexity in saddle-dominated scrambling
- Integrable and chaotic dynamics of spins coupled to an optical cavity
- Learning efficient decoders for quasi-chaotic quantum scramblers
- Transitions in Entanglement Complexity in Random Circuits
- Chaos in a classical limit of the Sachdev-Ye-Kitaev model
- Classical and quantum chaos in a three-mode bosonic system
- Fast scrambling without appealing to holographic duality
- Random Matrix Theory of the Isospectral twirling
- Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the Luttinger-liquid model
- Retrieving information from a black hole using quantum machine learning
- Quantum echo dynamics in the Sherrington-Kirkpatrick model
- From ETH to algebraic relaxation of OTOCs in systems with conserved quantities
- Angular momentum and chaos bound of charged particles around Einstein-Euler-Heisenberg AdS black holes
- Out-of-time-ordered correlators and the Loschmidt echo in the quantum kicked top: How low can we go?
- Refined quantum Lyapunov exponents from replica out-of-time-order correlators
- Surprises in the Deep Hilbert Space of all-to-all systems: From super-exponential scrambling to slow entanglement growth
- Complexity of quantum motion and quantum-classical correspondence: A phase-space approach
- Similar early growth of out-of-time-ordered correlators in quantum chaotic and integrable Ising chains
- Exact relaxation dynamics and quantum information scrambling in multiply quenched harmonic chains
- Witnessing quantum chaos using observational entropy
- Slow relaxation of out-of-time-ordered correlators in interacting integrable and nonintegrable spin-1/2 XYZ chains
- Quantum Information Scrambling in Molecules
- Quantum information scrambling in two-dimensional Bose-Hubbard lattices
- Classical and quantum butterfly effect in nonlinear vector mechanics
- Microscope for Quantum Dynamics with Planck Cell Resolution
- Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise
- Measuring out-of-time-ordered correlation functions with a single impurity qubit in a bosonic Josephson junction
- The Limits of Quantum Information Scrambling
- Probing chaos in the spherical p-spin glass model
- Emergence of a Renormalized Expansion in Quenched Critical Many-Body Systems
- Fast scrambling dynamics and many-body localization transition in an all-to-all disordered quantum spin model
- Resonant detectors of gravitational wave in the linear and quadratic generalized uncertainty principle framework
- Relaxation exponents of OTOCs and overlap with local Hamiltonians
- Out of Time Order Correlation of the Hubbard Model with Random Local Disorder
- Wigner current in multidimensional quantum billiards
- Quantum simulations of complex systems
- Out-of-Time-Order-Correlation in perturbed quantum wells
- False signals of chaos from quantum probes
- From single-particle to many-body chaos in Yukawa--SYK: theory and a cavity-QED proposal
- On a Matrix Ensemble for Arbitrary Complex Quantum Systems