Phenomenological Ginzburg-Landau theory for triple-Q magnetic orders on a hexagonal lattice

Complex magnets on hexagonal lattices frequently exhibit ordering at three symmetry-related wave vectors. Here, the authors develop a comprehensive Ginzburg-Landau theory that establishes a foundational framework for triple-Q magnetic orders in $O$(2) and $O$(3) models. They derive complete phase diagrams, revealing single-Q, double-Q, and triple-Q states---including collinear, orthogonal, and 120\ifmmode^\circ\else\textdegree\fi{} configurations---and analyze the associated collective excitations and symmetry breaking. This framework elucidates exotic magnetic orders in frustrated magnets such as Na${}_{2}$Co${}_{2}$TeO${}_{6}$.