Classification of point-group-symmetric orientational ordering tensors

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 $\{C_n,D_n,T,O,I\}\subset \text{SO}\left(3\right)$ and $\{C_{nv},S_{2n},C_{nh},D_{nh},D_{nd},T_h,T_d,O_h,I_h\}\subset \text{O}\left(3\right)$ for $n,2n\in \{1,2,3,4,6,\infty \}$. This central result may be perceived as a road map for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in, e.g., nanocolloidal systems and novel magnets.

Figure 1

A selection of finite subgroups of $\text{O}\left(3\right)$ or 3D point groups and their subgroup relations (not unique). They comprise continuous subgroups with a $C_{\infty }\simeq \text{SO}\left(2\right)$ axis and finite point groups. Ascending the hierarchy of symmetries, the order parameter tensor describing the phases associated with the point-group symmetries becomes increasingly complicated.