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

Motivated by the recent experiment on $\rm{K_2Cu_3O\left(SO_4\right)_3}$, an edge-shared tetrahedral spin-cluster compound [M. Fujihala \textit{et al.}, Phys. Rev. Lett. \textbf{120}, 077201 (2018)], we investigate two-leg spin-cluster ladders with the plaquette number $n_p$ in each cluster up to six by the density-matrix renormalization group method. We find that the phase diagram 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 the other 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 Haldane phase and an even Haldane phase. Moreover, in the latter case the region of the Haldane phase increases while the cluster-singlet phase and the even Haldane phase 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 the ground-state energy and entanglement entropy we obtain the order of the phase transtions. In particular, for $n_p=1$ there is no phase transition between the even Haldane phase and the cluster-singlet phase while for other odd $n_p$ there is a first-order phase transition. Our work provides comprehensive phase diagrams for these cluster-based models and may be helpful to understand experiments on related materials.