  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 anharmonic
oscillations (quantum anharmonic crystal), is considered. For this system, it
is proven that: (a) the set of tempered Gibbs measures is non-void and weakly
compact; (b) every Gibbs measure obeys an exponential integrability estimate,
the same for all such measures; (c) every Gibbs measure has a Lebowitz-Presutti
type support; (d) the set of all Gibbs measures is a singleton at high
temperatures. In the case of attractive interaction and one-dimensional
oscillations we prove that at low temperatures the system undergoes a phase
transition. 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 of all Gibbs
measures has been done, which essentially extends and refines the results
obtained so far for models of this type.
