Journal of Statistical Physics

, Volume 80, Issue 1–2, pp 223–271 | Cite as

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

  • Yong Moon Park
  • Hyun Jae Yoo


We consider quantum unbounded spin systems (lattice boson systems) in ν-dimensional lattice space Zν. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly β-dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.

Key Words

Quantum unbounded spin systems Wiener integral Gibbs states cluster expansion clustering property probability estimates 


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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Yong Moon Park
    • 1
  • Hyun Jae Yoo
    • 2
  1. 1.Department of Mathematics and Institute of Mathematical SciencesYonsei UniversitySeoulKorea
  2. 2.Department of PhysicsYonsei UniversitySeoulKorea

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