Unveiling Mysteries of GdRu$_2$Si$_2$: 3D Magnetism in a layered like Magnet

Abstract: GdRu$2$Si$_2$, a centrosymmetric magnet with a square lattice of Gd atoms, hosts a short-period skyrmion square lattice (SkL) without Dzyaloshinskii-Moriya interaction (DMI). RKKY-type exchange between the Gd moments results in an exchange frustration, which is the main source of non-collinearity in the spiral phases of the system. Previous studies focused on the in-plane $\vec{Q}$ vectors in understanding the magnetic phases of the system as they appear and have been observed on the 2D Gd layers. In this work, we calculate the Gd-Gd magnetic exchange interactions ($J{ij}$) and perform atomic spin dynamics (ASD) simulations, providing new insights about GdRu$2$Si$_2$. Our calculated $J{ij}$ shows that the strongest magnetic interaction occurs between Gd atoms along the [111] body-diagonal direction of the unit cell. This, along with the body-centered tetragonal structure of the Gd sublattice, points to the presence of a hitherto ignored modulation vector, $\vec{Q}{[111]}$, along the [111] direction in the spiral phases of the system. ASD simulations confirm this interlayer modulation, demonstrating that GdRu$_2$Si$_2$'s magnetic phases are more complex than suggested by 2D layer observations. The total magnetic order is determined by $\vec{Q}{[111]}$ alongside intralayer $\vec{Q}{[100]}$ and $\vec{Q}{[010]}$, establishing GdRu$_2$Si$_2$ as a strong 3D magnet requiring comprehensive theoretical modeling. Considering these, our ASD simulations accurately reproduce experimental phase transitions and highlight the significant role of dipolar interactions (due to the large Gd moment) over a weak uniaxial anisotropy in determining the ground state. This work enhances the understanding of GdRu$_2$Si$_2$'s complex magnetism, suggesting similar interlayer effects may be important in other layered magnetic systems.