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The paradigm of spontaneous symmetry breaking encompasses the breaking of therotational symmetries $O(3)$ of isotropic space to a discrete subgroup, i.e. athree-dimensional point group. The subgroups form a rich hierarchy and allowfor many different phases of matter with orientational order. Such spontaneoussymmetry breaking occurs in nematic liquid crystals and a highlight of suchanisotropic liquids are the uniaxial and biaxial nematics. Generalizing thefamiliar uniaxial and biaxial nematics to phases characterized by an arbitrarypoint group symmetry, referred to as \emph{generalized nematics}, leads to alarge hierarchy of phases and possible orientational phase transitions. Wediscuss how a particular class of nematic phase transitions related to axialpoint groups can be efficiently captured within a recently proposed gaugetheoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J.Slager, K. Wu, J. Zaanen, Phys. Rev. X {\bf 6}, 041025 (2016)]. Thesetransitions can be introduced in the model by considering anisotropic couplingsthat do not break any additional symmetries. By and large this generalizes thewell-known uniaxial-biaxial nematic phase transition to any arbitrary axialpoint group in three dimensions. We find in particular that the generalizedaxial transitions are distinguished by two types of phase diagrams withintermediate vestigial orientational phases and that the window of thevestigial phase is intimately related to the amount of symmetry of the definingpoint group due to inherently growing fluctuations of the order parameter. Thismight explain the stability of the observed uniaxial-biaxial phases as comparedto the yet to be observed other possible forms of generalized nematic orderwith higher point group symmetries.

Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematics