Fig. 1: Crystal structure and zero-field magnetic order of a Gd-based breathing kagomé lattice.

a Hexagonal unit cell of Gd₃Ru₄Al₁₂, where a, b, and c are crystallographic lattice directions. b Within the Gd₃Al₄ layer, rare earth (Gd) atoms form a distorted kagomé net with alternating distances r, r’ between nearest neighbors. Al and Ru atoms are not shown. The black rhombus indicates the size of the primitive unit cell. c Magnetic susceptibility (blue, left axis) increases continuously in the paramagnetic state as temperature is lowered. The inverse susceptibility H/M (red, right axis) is fitted by the Curie–Weiss expression (dashed line) at high temperature. d, e Specific heat c_P(T) and M/H show two phase transitions in zero magnetic field. f, g At the (7, 0, 0) + q₃ = (7 + q, −q, 0) incommensurate reflection, resonant x-ray scattering with polarization analysis provides modulated moments within (m_{⊥ q,c}, blue solid triangles) and perpendicular to (m_//c, red open triangles) the hexagonal plane, as well as the magnitude of the ordering vector q. Inset of (c) six directions of q_i are allowed by symmetry. The black hexagon indicates a conventional unit cell in real space. The transition temperatures T_N2 > T_N1 bound the red shaded area in (d–g).