05-323 Yuri Kozitsky and Tatiana Pasurek

Euclidean Gibbs Measures of Quantum Anharmonic Crystals (573K, PDF) Sep 16, 05

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Abstract. A lattice system of interacting temperature loops, which is used in the Euclidean approach to describe equilibrium thermodynamic properties of an infinite system of interacting quantum particles performing $\nu$-dimensional anharmonic oscillations (quantum anharmonic crystal), is considered. For this system, it is proven that: (a) the set of tempered Gibbs measures $\mathcal{G}^{\rm t}$ is non-void and weakly compact; (b) every $\mu \in \mathcal{G}^{\rm t}$ obeys an exponential integrability estimate, the same for the whole set $\mathcal{G}^{\rm t}$; (c) every $\mu \in \mathcal{G}^{\rm t}$ has a Lebowitz-Presutti type support; (d) $\mathcal{G}^{\rm t}$ is a singleton at high temperatures. In the case of attractive interaction and $\nu=1$ we prove that at low temperatures the system undergoes a phase transition, i.e., $|\mathcal{G}^{\rm t}|>1$. The uniqueness of Gibbs measures due to strong quantum effects (strong diffusivity) and at a nonzero external field are also proven in this case. Thereby, a complete description of the properties of the set $\mathcal{G}^{\rm t}$ has been done, which essentially extends and refines the results obtained so far for models of this type.

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