RKKY-dipolar Interactions and 3D Spin Supersolid on Stacked Triangular Lattice

Inspired by the recent discovery of metallic spin supersolidity and its giant magnetocaloric effect in the rare-earth alloy EuCo₂Al₉ 1, we perform a combined study through electronic structure analysis, effective spin model, and Monte Carlo simulations on a stacked triangular lattice, and reveal a novel mechanism for the emergence of 3D spin supersolid in a metallic antiferromagnet. From first-principles inputs, we derive a minimal spin model on a stacked triangular lattice (STL), which arises from the interplay between Ruderman-Kittel-Kasuya-Yosida (RKKY) and dipolar interactions and accurately reproduces the experimental thermodynamics. Based on the STL model, we identify a ground state that simultaneously breaks discrete lattice translational symmetry and continuous spin-rotational symmetry — the hallmark of a spin supersolid. Furthermore, we present the field-temperature phase diagram of the 3D STL model and discuss the various magnetic phases and associated phase transitions. Under zero field, the spin supersolid Y order establishes in two steps: an upper transition at T_N1, where an emergent U(1) symmetry appears and the system enters a fluctuating collinear regime, followed by a lower transition at T_N2 into the spin supersolid Y phase. In contrast, the supersolid V phase undergoes a single phase transition at T_N^V . Our results not only provide a comprehensive theoretical understanding of the metallic spin supersolid reported for EuCo₂Al₉ but also pave the way for further experimental investigations into its supersolid transitions and universality class.