In this paper, we present a phase classification of (effectively) two-dimensional non-Abelian ne
matics, obtained using the Hopf symmetry breaking for
malism. In this for
malism, one exploits the underlying double symmetry which treats both ordinary and topological modes on equal footing, i.e., as representations of a single (non-Abelian) Hopf symmetry. The method introduced in the literature [F.A. Bais, B.J. Schroers, J.K. Slingerland, Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 (2002) 181601; F.A. Bais, B.J. Schroers, J.K. Slingerland, Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory, JHEP 05 (2003) 068.] and further developed in a paper published in parallel [F.A. Bais, C.J.M. Mathy, The breaking of quantum double symmetries by defect condensation, 2006, arXiv:cond-
mat/06
02115.] allows for a full classification of defect mediated as well as ordinary symmetry breaking patterns and a description of the resulting confinement and/or liberation phenomena. After a sum
mary of the for
malism, we determine the double symmetries for tetrahedral, octahedral, and icosahedral ne
matics and their representations. Subsequently the breaking patterns which follow from the for
mation of admissible defect condensates are analyzed syste
matically. This leads to a host of new (quantum and classical) ne
matic phases. Our result consists of a listing of condensates, with the corresponding intermediate residual symmetry algebra
and the symmetry algebra characterizing the effective “low energy” theory of surviving unconfined and liberated degrees of freedom in the broken phase. The results suggest that the formalism is applicable to a wide variety of two-dimensional quantum fluids, crystals and liquid crystals.