Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms

TL;DR

This paper reviews and extends Lieb-Robinson bounds for quantum lattice systems, providing a unified framework for analyzing quasi-locality, including systems with infinite-dimensional spaces and unbounded Hamiltonians, with applications to phase stability.

Contribution

It introduces a generalized framework for quasi-local maps in quantum lattice systems, accommodating infinite-dimensional spaces and unbounded Hamiltonians, and sets the stage for analyzing phase stability.

Findings

Unified framework for quasi-locality in quantum systems

Extension to infinite-dimensional and unbounded Hamiltonian systems

Foundation for stability analysis of gapped ground states

Abstract

Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustration-free models satisfying a Local Topological Quantum Order condition, which we present in a sequel to this paper.