Spectral Gaps for Spin Systems: Some Non-convex Phase Examples

We prove that a convex phase may be perturbed into a non-convex phase preserving the spectral gap properties of the unbounded spin system with nearest neighbour interaction associated to this potential. The proof is based on Helfferʹs method hat reduces the spectral properties of the unbounded spin system to some uniform spectral gap of the one-dimensional phase. We then make use of Hardyʹs criterion for Poincaré inequalities on the real line to construct our examples.