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

Sven Bachmann, Wojciech Dybalski, Pieter Naaijkens
December 09, 2014
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. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.