Lieb–Robinson Bounds, Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems

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.

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Correspondence to Wojciech Dybalski.

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Communicated by Claude Alain Pillet.

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Bachmann, S., Dybalski, W. & Naaijkens, P. Lieb–Robinson Bounds, Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems. Ann. Henri Poincaré 17, 1737–1791 (2016). https://doi.org/10.1007/s00023-015-0440-y

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Keywords

  • Ising Model
  • Mass Shell
  • Local Observable
  • Scatter Theory
  • Asymptotic Completeness