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An S = 1 2 three-leg antiferromagnetic spin nanotube is investigated using the DMRG method and the level spectroscopy analysis combined with the numerical exact diagonalization. It is known that the system has a spin gap, when the three rung couplings are equivalent. In this paper, we see that as one of the three rung exchanges varies, a quantum phase transition occurs at a critical value of the varied rung exchange, and then the system is in a gapless Tomonaga–Luttinger-liquid phase. A ground-state phase diagram containing this phase transition is obtained by the phenomenological renormalization.

Using numerical methods, we study the finite-field ground state of an S = 1 zigzag ladder. Special attention is paid to the magnetization plateau which appears at a half of the saturation magnetization in the ground-state magnetization curve.

We investigate the ground state phase diagram of the S = 1 2 distorted diamond type spin chain with the XXZ anisotropy by use of the degenerate perturbation theory and the level spectroscopy analysis of the numerical diagonalization data for the finite-size systems. As the XXZ anisotropy Δ is increased from 0 to ∞ , there appears the reentrant quantum phase transition TL–SF ⇒ Neel ⇒ TL–SF ⇒ ferrimagnetic, when the coupling parameters have suitable values. Here TL–SF denotes the Tomonaga–Luttinger spin-fluid state. Further we find that four Berezinskii–Kosterlitz–Thouless lines meet together at one point in the phase diagram on the Δ versus J 3 / J 2 plane. These interesting phenomena are attributed to the interplay between the trimer nature, frustrating interactions and the XXZ anisotropy. Similar phenomena can be found in several systems having these three keywords.