  The quest for a successful condensed matter theory that incorporates
diffusion of cations, whose trajectories are restricted to a
honeycomb/hexagonal pattern prevalent in honeycomb layered materials is
ongoing, with the recent progress discussed herein focusing on symmetries,
topological aspects and phase transition descriptions of the theory. Such a
theory is expected to differ both qualitatively and quantitatively from 2D
electron theory on static carbon lattices, by virtue of the dynamical nature of
diffusing cations within lattices in honeycomb layered materials. Herein, we
have focused on recent theoretical progress in the characterisation of
pnictogen- and chalcogen-based honeycomb layered oxides with emphasis on
hexagonal/honeycomb lattices of cations. Particularly, we discuss the link
between Liouville conformal field theory to expected experimental results
characterising the optimal nature of the honeycomb/hexagonal lattices in
congruent sphere packing problems. The diffusion and topological aspects are
captured by an idealised model, which successfully incorporates the duality
between the theory of cations and their vacancies. Moreover, the rather
intriguing experimental result that a wide class of silver-based layered
materials form stable Ag bilayers, each comprising a pair of triangular
sub-lattices, suggests a bifurcation mechanism for the Ag triangular
sub-lattices, which ultimately requires conformal symmetry breaking within the
context of the idealised model, resulting in a cation monolayer-bilayer phase
transition. Other relevant experimental, theoretical and computational
techniques applicable to the characterisation of honeycomb layered materials
have been availed for completeness.
