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Motivated by the recent experiment on $\rm{K_2Cu_3O\left(SO_4\right)_3}$, anedge-shared tetrahedral spin-cluster compound [M. Fujihala \textit{et al.},Phys. Rev. Lett. \textbf{120}, 077201 (2018)], we investigate two-legspin-cluster ladders with the plaquette number $n_p$ in each cluster up to sixby the density-matrix renormalization group method. We find that the phasediagram of such ladders strongly depends on the parity of $n_p$. For even$n_p$, the phase diagram has two phases, one is the Haldane phase, and theother is the cluster rung-singlet phase. For odd $n_p$, there are four phases,which are a cluster-singlet phase, a cluster rung-singlet phase, a Haldanephase and an even Haldane phase. Moreover, in the latter case the region of theHaldane phase increases while the cluster-singlet phase and the even Haldanephase shrink as $n_p$ increases. We thus conjecture that in the large $n_p$limit, the phase diagram will become independent of $n_p$. By analysing theground-state energy and entanglement entropy we obtain the order of the phasetranstions. In particular, for $n_p=1$ there is no phase transition between theeven Haldane phase and the cluster-singlet phase while for other odd $n_p$there is a first-order phase transition. Our work provides comprehensive phasediagrams for these cluster-based models and may be helpful to understandexperiments on related materials.

Parity-dependent phase diagrams in spin-cluster two-leg ladders