Abstract
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the “reservoir” that facilitates the dissipation. In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
- Received 12 December 2017
DOI:https://doi.org/10.1103/PhysRevX.8.031057
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
At the microscopic level, an ensemble of many interacting quantum particles perfectly isolated from its environment can undergo reversible dynamics. Yet irreversible macroscopic behavior such as diffusion may emerge from the same system. How irreversible macroscopic dynamics can arise from reversible microscopic dynamics is a nontrivial question for both classical and quantum mechanical systems. Here, we present a model quantum system that explicitly illustrates how diffusion, and hence dissipation, can emerge.
We use a random quantum circuit model that is constrained to have a conservation law. This model consists of an array of quantum spins undergoing unitary dynamics with a discrete time. On each time step, nearest-neighbor pairs of spins interact quantum mechanically via a unitary quantum gate specific to that pair of spins at that time step. The action of each gate conserves one component of the total spin but is otherwise fully random. This randomness allows explicit analytic calculation of many correlation functions for this system.
We find that simple observable operators, such as the local density of the conserved spin component, are converted to less local operators. This produces a flow of operators to highly nonlocal, and thus essentially nonobservable, operators. This is the mechanism by which this system produces a dissipative bath for itself.
Our approach could help in understanding the many open questions about thermalization and quantum statistical mechanics in closed systems with conservation laws.
