Classification of point-group-symmetric orientational ordering tensors

Abstract

The concept of symmetry breaking has been a propelling force in understanding phases of matter. While rotational symmetry breaking is one of the most prevalent examples, the rich landscape of orientational orders breaking the rotational symmetries of isotropic space, i.e. $O(3)$, to a three-dimensional point group remain largely unexplored, apart from simple examples such as ferromagnetic or uniaxial nematic ordering. Here we provide an explicit construction, utilizing a recently introduced gauge theoretical framework, to address the three-dimensional point-group-symmetric orientational orders on a general footing. This unified approach allows us to enlist order parameter tensors for all three dimensional point groups. By construction, these tensor order parameters are the minimal set of simplest tensors allowed by the symmetries that uniquely characterize the orientational order. We explicitly give these for the point groups $\{C_n,D_n,T,O,I\}\subset SO(3)$ and $\{C_{nv},S_n,C_{nh},D_{nh},D_{nd},T_h,T_d,O_h,I_h\}\subset O(3)$ for $n=\{1,2,3,4,6,\infty \}$. This central result may be perceived as a roadmap for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in e.g. nano-colloidal systems and novel magnets.

14 pages, 1 figure, 1 table; published version in Physical Review E 2016

References in corpus (4)

• Disclinations, dislocations and continuous defects: a reappraisal
• Classifying Novel Phases of Spinor Atoms
• Landau-deGennes Theory of Biaxial Nematics Re-examined
• Classification of nematic order in 2+1D: Dislocation melting and $O(2)/Z_N$ lattice gauge theory

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