Communications in Mathematical Physics

, Volume 144, Issue 3, pp 443–490

Finitely correlated states on quantum spin chains

  • M. Fannes
  • B. Nachtergaele
  • R. F. Werner

DOI: 10.1007/BF02099178

Cite this article as:
Fannes, M., Nachtergaele, B. & Werner, R.F. Commun.Math. Phys. (1992) 144: 443. doi:10.1007/BF02099178


We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Néel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • M. Fannes
    • 1
    • 2
  • B. Nachtergaele
    • 3
    • 4
  • R. F. Werner
    • 6
  1. 1.Inst. Theor. FysicaUniversiteit LeuvenLeuvenBelgium
  2. 2.Bevoegdverklaard NavorserN.F.W.O.Belgium
  3. 3.Depto de FísicaUniversidad de ChileSantiago de Chile
  4. 4.Onderzoeker I.I.K.W.Belgium
  5. 5.Universiteit LeuvenBelgium
  6. 6.Dublin Institute for Advanced StudiesDublin 4Ireland
  7. 7.Universität OsnabrückFRG

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