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We consider translation invariant gapped quantum spin systems satisfying theLieb-Robinson bound and containing single-particle states in a ground staterepresentation. Following the Haag-Ruelle approach from relativistic quantumfield theory, we construct states describing collisions of several particles,and define the corresponding $S$-matrix. We also obtain some generalrestrictions on the shape of the energy-momentum spectrum. For the purpose ofour analysis we adapt the concepts of almost local observables andenergy-momentum transfer (or Arveson spectrum) from relativistic QFT to thelattice setting. The Lieb-Robinson bound, which is the crucial substitute ofstrict locality from relativistic QFT, underlies all our constructions. Ourresults hold, in particular, in the Ising model in strong transverse magneticfields.

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