Generalized Liquid Crystals: Giant Fluctuations and the Vestigial Chiral Order ofI,O, andTMatter

Abstract

The physics of nematic liquid crystals has been the subject of intensive research since the late 19th century. However, the focus of this pursuit has been centered around uniaxial and biaxial nematics associated with constituents bearing a D ∞h or D 2h symmetry, respectively. In view of general symmetries, however, these are singularly special since nematic order can in principle involve any point-group symmetry. Given the progress in tailoring nanoparticles with particular shapes and interactions, this vast family of "generalized nematics" might become accessible in the laboratory. Little is known because the order parameter theories associated with the highly symmetric point groups are remarkably complicated, involving tensor order parameters of high rank. Here, we show that the generic features of the statistical physics of such systems can be studied in a highly flexible and efficient fashion using a mathematical tool borrowed from high-energy physics: discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice gauge models encapsulating nematic ordering of general three-dimensional point-group symmetries. We find that the most symmetrical generalized nematics are subjected to thermal fluctuations of unprecedented severity. As a result, novel forms of fluctuation phenomena become possible. In particular, we demonstrate that a vestigial phase carrying no more than chiral order becomes ubiquitous departing from high point-group symmetry chiral building blocks, such as I, O, and T symmetric matter.

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].…”

“…It was subsequently found that discrete, non-Abelian gauge theory can be mobilized to compute both the explicit order parameters as well as the generic statistical physics associated with this symmetry breaking in a relatively straightforward way. With regard to the latter, it was found that in case of the most symmetric point groups one runs into thermal fluctuation effects of an unprecedented magnitude [59]. In the present context we just ignore these complications.…”

“…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.…”

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].…”