Ground States and Critical Points for Aubry-Mather Theory in Statistical Mechanics

We consider statistical mechanics systems defined on a set with some symmetry properties (namely, the set admits an action by a group, which is finitely generated and residually finite). Each of the sites has a real order parameter. We assume that the interaction is ferromagnetic as well as symmetric with respect to the action of the symmetry group as well as invariant under the addition of an integer to the order parameter of all the sites. Given any cocycle of the symmetry group, we prove that there are ground states which satisfy an order condition (known as Birkhoff property) and that are at a bounded distance to the cocycle. The set of ground states we construct for a cocyle, is ordered. Equivalently, this set of ground states forms a (possibly singular) lamination. Furthermore, we show that, given any completely irrational cocycle, either the above lamination consists of a foliation made of a continuous one parameter family of ground states, or, inside any gap of the lamination, ...

• Type: Journal
• Year: 2010
• Where: JNS
• Authors: Rafael de La Llave, Enrico Valdinoci
• Keywords: Ground State, JNS 2010, Order Parameter, Symmetry Group