Locality of dynamics in general harmonic quantum systems

TL;DR

This paper extends Lieb-Robinson bounds to harmonic quantum systems with infinite-dimensional constituents, analyzing how locality in dynamics depends on the nature of interactions and applying results to quantum field models.

Contribution

It formulates Lieb-Robinson bounds for general harmonic systems on lattices, including non-local interactions, and explores their implications for quantum field models and non-equilibrium dynamics.

Findings

Local interactions lead to exponentially suppressed corrections to locality.

Non-local algebraic interactions cause algebraic decay of correlations.

In the continuum limit, lattice locality results recover exact causality in quantum fields.

Abstract

The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time under a local Hamiltonian will essentially grow linearly in their support, up to exponentially suppressed corrections. In this work, we formulate Lieb-Robinson bounds for general harmonic systems on general lattices, for which the constituents are infinite-dimensional, as systems representing discrete versions of free fields or the harmonic approximation to the Bose-Hubbard model. We consider both local interactions as well as infinite-ranged interactions, showing how corrections to locality are inherited from the locality of the Hamiltonian: Local interactions result in stronger than exponentially suppressed corrections, while non-local algebraic interactions result in algebraic suppression. We derive bounds for canonical operators, Weyl operators and outline generalization to arbitrary operators. As an example, we discuss the Klein-Gordon field, and see how the approximate locality in the lattice model becomes the exact causality in the field limit. We discuss the applicability of these results to quenched lattice systems far from equilibrium, and the dynamics of quantum phase transitions.