Yuri Kozitsky and Tatiana Pasurek
Euclidean Gibbs Measures of Quantum Anharmonic Crystals
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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.