Fig. 1: Distorted kagome lattice and classical \(\overrightarrow{Q}=(1/3,1/3)\) magnetic order.
From: Phase diagram of a distorted kagome antiferromagnet and application to Y-kapellasite
a Schematic illustration of the three exchange couplings characterizing the effective Heisenberg Hamiltonian for Y-kapellasite (J⬡, J, and \(J^{\prime}\)) shown in red, blue, and green, respectively. The presence of three different couplings breaks the translational symmetry of the kagome lattice and leads to a decorated triangular lattice with an enlarged unit cell of nine sites, here represented by the black hexagon (Wigner–Seitz cell). The Hamiltonian of the system is periodic under translations along the Bravais vectors \({\overrightarrow{a}}_{1}\) and \({\overrightarrow{a}}_{2}\) and the sites within the unit cell are divided into sublattices A (hollow symbols) and B (solid symbols). Due to the different values of the three exchange terms, the D6 point group symmetry of the kagome lattice is broken down to C6. b Pictorial view of the reciprocal space. The blue arrows represent the unit vectors of the reciprocal space (\({\overrightarrow{b}}_{1}\) and \({\overrightarrow{b}}_{2}\)). The gray hexagons tiling the reciprocal space depict the first Brillouin zone of the lattice, while the black dashed hexagon delimits the so-called extended Brillouin zone. Some of the high symmetry points are marked with black dots. Finally, red lines represent the path along which the magnon dispersion is plotted in Fig. 3. c Classical \(\overrightarrow{Q}=(1/3,1/3)\) magnetic order for \(J \,>\, J^{\prime}\) (red region of Fig. 2). The orientations of the spins are fully specified by the angle ϕ between neighboring spins in the J⬡-hexagons, as outlined in the main text (here we take the value of ϕ for the case \(J^{\prime} =0\) and J⬡ = J). In this figure, the spins are arranged in the xy-plane and their orientation is represented by the angle with respect to the Sx axis. The red, blue, and green colors of the spins of sublattice A help visualizing the \(\overrightarrow{Q}=(1/3,1/3)\) pattern. d The classical \(\overrightarrow{Q}=(1/3,1/3)\) magnetic order of Fig. 1c in the J ≫ J⬡ limit (\(J^{\prime} =0\)). The spins form antiferromagnetic trimers along the J-bonds (depicted in black). The trimers are arranged in an effective kagome lattice structure and their orientations, highlighted by the three different colors, follow the \(\sqrt{3}\times \sqrt{3}\) pattern69.
