Abstract
Thermalization and scrambling are the subject of much recent study from the perspective of many-body quantum systems with locally bounded Hilbert spaces (“spin chains”), quantum field theory, and holography. We tackle this problem in 1D spin chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs) and entanglement growth in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a “hydrodynamical” equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture, quantum information travels in a front with a “butterfly velocity” that is smaller than the light-cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do not observe a prolonged exponential regime of the form for a fixed Lyapunov exponent . We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic Floquet ergodic systems, and we support this idea by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional nonrandom Floquet spin chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization.
- Received 17 July 2017
DOI:https://doi.org/10.1103/PhysRevX.8.021013
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
After a long period of time, physical systems tend to reach a thermal equilibrium where the final state is well described by a small number of parameters, such as temperature or pressure. At this point, the details of the initial conditions are “scrambled” and can no longer be determined by simple local measurements. It is of particular interest how such equilibration occurs on the atomic scale in quantum systems of many interacting particles, where scrambling is related to the buildup of delicate quantum correlations and entanglement. In this work, we uncover some hitherto unknown universal features of the dynamics of information in one dimension, showing that quantum information flows and diffuses according to a simple “hydrodynamical” rule encountered in the study of fluids.
We present analytical results for a toy model, given by a circuit of randomly chosen local unitary operations. We give an exact coarse-grained description of how observables evolve in time, and we relate this quantity to so-called out-of-time-order correlation functions, which form the focus of recent studies in holography and quantum chaos. We also relate operator evolution to entanglement growth and determine the entanglement velocity (or growth rate) analytically.
While our exact results concern average quantities in the random circuit model, we propose that they represent generic features of certain one-dimensional many-body quantum systems. We support this conjecture with extensive numerical calculations, showing that similar diffusive operator-spreading behavior appears in a family of clean, periodically driven spin chains, which are the focus of current experimental efforts in cold atomic gases.
