Classification of point-group-symmetric orientational ordering tensors

TL;DR

This paper develops a comprehensive framework to classify and construct order parameter tensors for all three-dimensional point-group-symmetric orientational orders, advancing understanding of symmetry-breaking in complex phases.

Contribution

It introduces a unified gauge theoretical approach to explicitly construct minimal tensor order parameters for all 3D point groups, enabling systematic identification of exotic orientational orders.

Findings

Constructed tensor order parameters for all 3D point groups.

Provided explicit forms for key symmetry groups such as Cn, Dn, T, O, I.

Facilitated the exploration of complex orientational phases in experimental systems.

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.