Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky, and Michael Roeckner
Euclidean Gibbs States of Quantum Lattice Systems
(211K, AMS-LATEX)
ABSTRACT. An approach to the description of the Gibbs states of lattice models of
interacting quantum anharmonic oscillators, based on integration in infinite
dimensional spaces, is described in a systematic way. Its main feature is the
representation of the local Gibbs states by means of certain probability
measures (local Euclidean Gibbs measures). This makes possible to employ the
machinery of conditional probability distributions, known in classical
statistical physics, and to define the Gibbs state of the whole system as a
solution of the equilibrium (Dobrushin-Lanford-Ruelle) equation. With the help
of this representation the Gibbs states are extended to a certain class of
unbounded multiplication operators, which includes the order parameter and the
fluctuation operators describing the long range ordering and the critical point
respectively. It is shown that the local Gibbs states converge, when the mass of
the particle tends to infinity, to the states of the corresponding classical
model. A lattice approximation technique, which allows one to prove for the
local Gibbs states analogs of known correlation
inequalities, is developed. As a result, certain new
inequalities are derived. By means of them, a number of results describing
physical properties of the model are obtained. Among them are: the existence of
the long-range order for low temperatures and large values of the particle's
mass; the suppression of the critical point behaviour for small values of the
mass and for all temperatures; the uniqueness of the Euclidean Gibbs states for
all temperatures and for the values of the mass less than a certain threshold
value, dependent on the temperature.