Title: Magnetic structure and component-separated transitions of HoNiSi$_{3}$

Abstract: HoNiSi$_{3}$ is an intermetallic compound characterized by two successive antiferromagnetic transitions at $T_{N1} = 6.3$ K and $T_{N2} = 10.4$ K. Here, its zero-field microscopic magnetic structure is inferred from resonant x-ray magnetic diffraction experiments on a single crystalline sample that complement previous bulk magnetic susceptibility data. For $T < T_{N2}$, the primitive magnetic unit cell matches the chemical cell. The magnetic structure features ferromagnetic {\it ac} planes stacked in an antiferromagnetic $\uparrow \downarrow \uparrow \downarrow$ pattern. For $T_{N1} < T < T_{N2}$, the ordered magnetic moment points along $\vec{a}$, and for $T < T_{N1}$ a component along $\vec{c}$ also orders. A symmetry analysis indicates that the magnetic structure for $T<T_{N1}$ is not compatible with the presumed orthorhombic $Cmmm$ space group of the chemical structure, and therefore a slight lattice distortion is implied. Mean-field calculations using a simplified magnetic Hamiltonian, including a reduced set of three independent exchange coupling parameters determined by density functional theory calculations and two crystal electric field terms taken as free-fitting parameters, are able to reproduce the main experimental observations. An alternative approach using a more complete model including seven exchange coupling and nine crystal electric field terms is also explored, where the search of the ground state magnetic structure compatible with the available anisotropic magnetic susceptibility and magnetization data is carried out with the help of an unsupervised machine learning algorithm. The possible magnetic configurations are grouped into five clusters, and the cluster that yields the best comparison with the experimental macroscopic data contains the parameters previously found with the simplified model and also predicts the correct ground-state magnetic structure.