A Hopf symmetry approach to the physics of ordinary and liquid crystals

We analyze the modes (excitations) of phases that arise via spontaneous global symmetry breaking from the isotropic phase. We focus on ordinary and liquid crystals. The modes can be classified into three categories: regular or electric modes, which refer to regular excitations of the ground state; topological or magnetic modes, which have a core in which the symmetry is (partially) restored; and dyons, which carry both electric and magnetic quantum numbers. We study their interactions, in particular their fusion and braiding properties. These interactions can be captured by a mathematical structure called a Hopf algebra, which we analyze in detail. Once the Hopf symmetry of a phase has been established, we study symmetry breaking induced by the condensation of electric, magnetic or dyonic modes. The use of Hopf symmetry breaking to analyze the phase structure of media with topological excitations was pioneered by Bais, Schoers and Slingerland. In this thesis we use a novel more refined criterion to define the residual symmetry, which in certain cases leads to a richer analysis of the physics of the broken phase. Some of the excitations are confined, meaning that they cost an infinite amount of energy to create. These confined excitations may form unconfined ”hadronic” composites, in analogy with the hadrons in the theory of strong interactions of quarks. We determine the unconfined symmetry algebra of the broken phase, and develop a method to analyze the hadronic composites in condensed matter systems. We apply our symmetry breaking analysis to ordinary and liquid crystals, and predict a wealth of new phases, in particular phases resulting from nonabelian defect condensates.