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In this work, which is based on spin-2 vectors and traceless spin-2 tensors,an effective Hamiltonian is constructed with a linearly dispersive five-folddegenerate point with spin-2 vector-momentum couplings. For the model withoutspin-2 vector-tensor coupling, the topological Chern numbers of five bands arecalculated as 4, 2, 0, -2, -4. After including spin-2 vector-tensor coupling,separate topological Chern numbers are obtained for different couplings. Acubic lattice of atoms with five internal states is designed to realize twofive-fold degenerate points. The Chern numbers of the bands can be changed bytuning the coupling coefficient. In this work we propose a theoretical designto obtain spin-2 quasi-particles.

Effective Spin-2 Quasi-particles at Linear Dispersive Five-fold Degenerate Points with Tunable Topological Chern Numbers