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The concept of symmetry breaking has been a propelling force in understandingphases of matter. While rotational symmetry breaking is one of the mostprevalent examples, the rich landscape of orientational orders breaking therotational symmetries of isotropic space, i.e. $O(3)$, to a three-dimensionalpoint group remain largely unexplored, apart from simple examples such asferromagnetic or uniaxial nematic ordering. Here we provide an explicitconstruction, utilizing a recently introduced gauge theoretical framework, toaddress the three-dimensional point-group-symmetric orientational orders on ageneral footing. This unified approach allows us to enlist order parametertensors for all three dimensional point groups. By construction, these tensororder parameters are the minimal set of simplest tensors allowed by thesymmetries that uniquely characterize the orientational order. We explicitlygive 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 roadmapfor identifying exotic orientational orders that may become more and more inreach in view of rapid experimental progress in e.g. nano-colloidal systems andnovel magnets.

Classification of point-group-symmetric orientational ordering tensors