Breaking of Euclidean Symmetry with an Application to the Theory of Crystallization

Abstract

We present a systematic study of the processes by which the original Euclidean invariance of a quantum statistical theory can be broken to produce pure phases with a lower symmetry. Our results provide a rigorous basis for Landau's argument on the nonexistence of critical point in the liquid‐solid phase transition. A classification of the possible residual symmetries is obtained, and its connection with spectral and cluster properties is established. Our tools are those of the algebraic approach to statistical mechanics; in particular, we make an extensive use of the KMS condition. None of our proofs involves the separability of the algebra of quasilocal observables.