Abstract
In certain analytically tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables reduces to an effective theory of an “entanglement membrane” in spacetime. These tractable systems involve an average over random local unitaries defining the dynamical evolution. We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries. Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits. This viewpoint provides a framework for applying ideas of coarse graining to dynamical quantities in chaotic systems. We apply the approach to some translationally invariant Floquet spin chains studied in the literature. We show that a consistent line tension may be defined for the entanglement membrane and that there are qualitative differences in this tension between generic models and “dual-unitary” circuits. These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to nonrandom Floquet models. We also provide an efficient numerical algorithm for determining the entanglement line tension in .
14 More- Received 7 January 2020
- Revised 6 June 2020
- Accepted 14 July 2020
DOI:https://doi.org/10.1103/PhysRevX.10.031066
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
Statistical physics is a lens on the microscopic world. At its highest resolution, an inconceivable number of particles interact via the laws of quantum mechanics. On zooming out to the macroscopic level, this daunting throng can oftentimes be described by a simple classical theory. This is the power of “coarse graining.” We use this lens to inspect the flow of quantum information and find that an unconventional hydrodynamic theory of membranelike objects emerges. In this work, we make sense of such membranes in generic quantum circuits.
To detect information spreading, we construct an interference protocol in which quantum circuits are duplicated an even number of times. A constructive interference phenomenon forces these copies to form pairs. The membrane—a line bisecting a circuit in a way that optimizes information flow across it—marks the boundary separating different ways to pair among the duplicated circuits. Imperfections slightly thicken the membrane, but a redefinition of the flow rate can account for it after coarse graining to macroscopic scales. We derive hydrodynamic equations to solve the flow rate.
The power of the coarse graining lies in the fact that only macroscopic data are needed to determine quantum information spreading. The membrane picture can have many extensions—for instance, in understanding a phase transition point beyond which information stops flowing. Intriguingly, the membrane has close connections to quantum gravity, where a membranelike object encircling a black hole measures the quantum information the black hole has engulfed. In the future, it may be possible to probe the properties of the membrane in many-particle quantum systems in the laboratory.