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Fedotovite K$_2$Cu$_3$O(SO$_4$)$_3$ is a candidate of new quantum spinsystems, in which the edge-shared tetrahedral (EST) spin-clusters consisting ofCu$^{2+}$ are connected by weak inter-cluster couplings to from one-dimensionalarray. Comprehensive experimental studies by magnetic susceptibility,magnetization, heat capacity, and inelastic neutron scattering measurementsreveal the presence of an effective $S$ = 1 Haldane state below $T \cong 4$ K.Rigorous theoretical studies provide an insight into the magnetic state ofK$_2$Cu$_3$O(SO$_4$)$_3$: an EST cluster makes a triplet in the ground stateand one-dimensional chain of the EST induces a cluster-based Haldane state. Wepredict that the cluster-based Haldene state emerges whenever the number oftetrahedra in the EST is $even$.

Cluster-based Haldane state in edge-shared tetrahedral spin-cluster chain: Fedotovite K$_2$Cu$_3$O(SO$_4$)$_3$