Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems

TL;DR

This paper extends scattering theory and spectral analysis to gapped quantum spin systems using Lieb-Robinson bounds, adapting relativistic QFT concepts to lattice models, and constructs a framework for understanding particle collisions and energy-momentum spectra.

Contribution

It develops a Haag-Ruelle type scattering theory for lattice spin systems, utilizing Lieb-Robinson bounds and Arveson spectrum, providing new insights into particle interactions in non-relativistic quantum models.

Findings

Constructed multi-particle collision states in gapped quantum spin systems.

Defined an $S$-matrix for lattice models using adapted Haag-Ruelle methods.

Derived restrictions on the energy-momentum spectrum shape.

Abstract

We consider translation invariant gapped quantum spin systems satisfying the Lieb-Robinson bound and containing single-particle states in a ground state representation. Following the Haag-Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding $S$-matrix. We also obtain some general restrictions on the shape of the energy-momentum spectrum. For the purpose of our analysis we adapt the concepts of almost local observables and energy-momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb-Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.