Motivated by the recent discovery of the spin tube $\left[{\left(\mathrm{Cu}{\mathrm{Cl}}_2\mathrm{tach}\mathrm{H}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$, we investigate the properties of a frustrated three-leg spin tube with antiferromagnetic intra- and inter-ring couplings. We pay special attention to the evolution of the properties from weak to strong inter-ring coupling and show on the basis of extensive density matrix renormalization group and exact diagonalization calculations that the system undergoes a first-order phase transition between a dimerized gapped phase at weak coupling that can be described by the usual spin-chirality model and a gapless critical phase at strong coupling that can be described by an effective spin-$3∕2$ model. We also show that there is a magnetization plateau at $1∕3$ in the whole gapped phase and slightly beyond. The implications for $\left[{\left(\mathrm{Cu}{\mathrm{Cl}}_2\mathrm{tach}\mathrm{H}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$ are discussed, with the conclusion that this system behaves essentially as a spin-$3∕2$ chain.

• Received 14 September 2005

DOI:https://doi.org/10.1103/PhysRevB.73.014409