Superspace Symmetry and Thermodynamics

Abstract

A thermodynamic criterion is proposed for a generalized symmetry operation in condensed matter to be well defined. It is basically the requirement that the spontaneous break of the defined symmetry should imply a phase transition. This criterion is applied to the rotational symmetry in the diffraction diagram of a quasi-periodic system, showing that the algebraic structure of the superspace groups is directly derivable from this physical requirement. Superspace symmetry, as usually defined in quasi-periodic systems, is the only structural geometrical property yielding rotational diffraction symmetry that is “stable” in the sense that it can only be broken if a phase transition takes place. In contrast, some geometrical configurations producing rotational diffraction invariances and known as “diffraction enhancement of symmetry” do not fulfil this stability criterion and are expected to happen only fortuitously. A physical necessary condition for any generalized symmetry in a condensed system is finally conjectured.