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 } ⊂ SO(3) andfor n,2n ∈ {1,2,3,4,6,∞}. 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.

Dual gauge field theory of quantum liquid crystals in three dimensions

“…In two dimensions, all rotational proper point groups are Abelian, while in 3D, the point groups are generally non-Abelian. As a consequence the order parameter theory of these 3D generalized nematics is a very rich and complex affair [59][60][61]. The uniaxial nematic has the pointgroup symmetry D ∞h , which breaks only two out of three rotational symmetries and the proper rotational part of which is Abelian; it is therefore not a good representative of rotational symmetry breaking in three dimensions.…”

“…Correcting the naive treatment of the KT papers [6,7], the classical theory of dislocation-mediated melting in 2D was established in the late 1970s by Nelson, Halperin, and Young [8][9][10] so that we now speak of the KTNHY-transition of a 2D crystal to a 2D liquid crystal. This famously includes the prediction of the hexatic phase, which we refer to as C 6 nematic [41,42,[59][60][61]. It was also realized that unbinding dislocations but not disclinations leads to the liquid crystal, while proliferation of dislocations and disclinations at the same time is in fact the ordinary first-order solid-liquid transition [32].…”

“…In fact, inspired by the considerations in the previous paragraph some of the authors felt a need to understand better the order parameter theory of such generalized (beyond uniaxial) nematics [59][60][61]. They found out that a systematic classification is just missing in the soft-matter literature, actually for a good reason.…”

“…As we discussed in Sec. II, the order-parameter theory already in the classical case becomes a quite complicated affair when dealing with anything else than the simple uniaxial nematics found in LCD screens and so forth [41,[59][60][61]. Here we will just avoid these difficulties by asserting that our quantum nematics are isotropic, which means "indistinguishable from the full spontaneous symmetry breaking of O(3) itself" when it comes to observable properties like the velocities of the rotational Goldstone bosons, or more generally the elasticity theory of the nematic [63].…”

Dual gauge field theory of quantum liquid crystals in two dimensions

“…5 comes to full fruition: one just generalizes the O(2)/Z N theory to O(3)/P where P is any of the 3D point groups and by taking the limit of strong gauge coupling it turns into a generating functional for the order parameter theories of the generalized nematics. We are presently exploring this fascinating landscape [193,194,195].…”