Euclidean Gibbs measures of interacting quantum anharmonic oscillators (Q996841)

This paper deals with equilibrium statistical mechanics of interacting \(\nu\)-dimensional quantum anharmonic oscillators index by a countable set \({\mathbb L}\). Usually Gibbs states are described by positive normalized functionals on the algebras of observables satisfying the KMS condition. However this approach does not work in the present model. The authors adapt the Euclidean approach. That is: The Feynman-Kac formula with \(\beta\)-periodic Ornstein-Uhlenbeck process yields a description of the model as the classical lattice system of infinite dimensional spins \( \omega_{\ell}: [0, \beta] \to {\mathbb R}^{\nu}\), \(\omega_{\ell}(0) = \omega_{\ell}(\beta)\), \(\ell \in {\mathbb L}\). Then, the DLR equation determines the set of Gibbs measures \({\mathcal G}^t\). The paper gives comprehensive description of \({\mathcal G}^t\): (i) Under certain general conditions, it is proved that \({\mathcal G}^t\) is non-void and compact; an exponential integrability and a Lebowitz-Presutti type support for every element of \({\mathcal G}^t\); \({\mathcal G}^t\) is a singleton at high temperatures. (ii) Under additional conditions of \(\nu =1\) and ferromagnetic interaction, the existence of phase transition and the uniqueness of the Gibbs measure at a nonzero external field are proved.