Existence and a priori estimates for Euclidean Gibbs states

Abstract

We prove a priori estimates and, as a sequel, the existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon–Nikodým derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence results generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and $N$-particle interactions for $N\in \mathbb {N}\cup \{\infty \}$ are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should also be of their own interest in infinite-dimensional analysis.

Locations

• Transactions of the Moscow Mathematical Society - View - PDF