Generic hamiltonian on a hyperkagome lattice

Geometric frustration typically arises in magnetic systems where it is impossible to simultaneously minimize all interactions, most commonly on lattices containing triangular units. In such frustrated systems, fluctuation effects become enhanced, often suppressing conventional magnetic ordering and providing a fertile ground for realizing exotic quantum phases such as quantum spin liquids. In this context, the recently synthesized hyperkagome lattice—a three-dimensional network composed of corner-sharing triangles—serves as an excellent platform to explore such unconventional magnetic phenomena. Motivated by this, our study aims to investigate the classical spin model on the hyperkagome lattice from a theoretical perspective. We first classify the symmetry-allowed spin interactions based on the lattice symmetry, and then analyze the static spin structure factor using the Self-Consistent Gaussian Approximation (SCGA). This approach allows us to identify the characteristic features of the classical spin liquid phases and the influence of geometric frustration on the magnetic correlations.