Critical points inside the gaps of ground state laminations for some models in statistical mechanics

We consider models of interacting particles situated in the points of a discrete set Λ. The state of each particle is determined by a real variable. The particles are interacting with each other and we are interested in ground states and other critical points of the energy (metastable states). Under the assumption that the set Λ and the interaction are symmetric under the action of a group G-which satisfies some mild assumptions-, that the interaction is ferromagnetic, as well as periodic under addition of integers, and that it decays with the distance fast enough, it was shown in a previous paper that there are many ground states that satisfy an order property called self-conforming or Birkhoff. Under some slightly stronger assumptions all ground states satisfy this order property. Under the assumption that the interaction decays fast enough with the distance, we show that either the ground states form a one dimensional family or that there are other Birkhoff critical points which are not ground states, but lying inside the gaps left by ground states. This alternative happens if and only if a Peierls-Nabarro barrier vanishes. The main tool we use is a renormalized energy. In the particular case that the set Λ is a one dimensional lattice and that the interaction is just nearest neighbor, our result establishes Mather's criterion for the existence of invariant circles in twist mappings in terms of the vanishing of the Peierls-Nabarro barrier.