Yuri Kozitsky and Tatiana Pasurek
Euclidean Gibbs Measures of Quantum Anharmonic Crystals
(573K, PDF)

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.