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We study the antiferromagnetic spin exchange models with $S=1/2$ and S=1 on aone-dimensional tetrahedron chain by both analytical and numerical approaches.The system is shown to be effectively mapped to a decoupled spin chain in theregime of strong rung coupling, and a spin sawtooth lattice in the regime ofweak rung coupling with spin $2S$ on the top row and spin $S$ on the lower row.The ground state for the homogeneous tetrahedron chain is found to fall into inthe regime of strong rung coupling. As a result, the elementary excitation forthe spin-$ 1/2$ system is gapless whereas the excitation for the spin-1 systemhas a finite spin gap. With the aid of the exact diagonalization method, wedetermine the phase diagram numerically and find the existence of an additionalphase in the intermediate regime. This phase is doubly degenerate and ischaracterized by an alternating distribution of rung singlet and rung spin$2S$. We also show that the SU(3) exchange model on the same lattice hascompletely different kind of ground state from that of its SU(2) correspondenceand calculate its ground state and elementary excitation analytically.Comment: 8 pages, 7 figures, version published in Phys. Rev.

Exact ground state and elementary excitations of the spin tetrahedron chain