Crystalline Stability and Order in One and Two Dimensions

Abstract

It is shown that no quantum system of particles, regardless of their interactions, can form a stable solid in one or two dimensions, if both the temperature and the compressibility are different from zero. By stable solid we mean that $〈u^2〉$ be finite. It is also proven that a neutral system of electrons and nuclei cannot exhibit crystalline order, of the usual kind, in one or two dimensions. This last proof is based on Bogolyubov's inequality.