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Quantum Phase Transition in the Twisted Three-Leg Spin Tube

We investigate the spin-1/2 twisted three-leg antiferromagnetic quantum spin tube in the presence of the easy-plane anisotropy, using the numerical diagonalization of finite-size clusters. And we observe cases of the existence and the absence of magnetization plateau at 1/3 height of saturation magnetization when the anisotropy is tuned. The phenomenological-renormalization analysis indicates a quantum phase transition between the 1/3 magnetization plateau phase and the plateauless one. The phase diagram is also presented.


Introduction
The spin system of a tube-type structure has attracted a lot of interest in the field of the molecule-based magnets. Particularly the spin-1/2 three-leg spin tube is the most important because it has strong frustration and quantum fluctuation [1]. One of interesting features is that regular tube has a spin gap, while the spin-1/2 three-leg spin ladder is gapless [2]. Motivated by the recent discovery of the spin tube [(CuCl 2 tachH) 3 Cl]Cl 2 (tach = cis,trans-1,3,5-triamino-cyclohexane) [3], we investigate the spin-1/2 twisted three-leg spin tube [4].
In the classical picture, the magnetization of the magnetic material is continuously increased with increasing external fields. On the other hand, a magnetic-field-induced spin gap is sometimes predicted to appear because of the quantum effects. It should appear as a plateau of the magnetization curve, namely a magnetization plateau or quantization of magnatization [5,6]. The numerical diagonalization study [5] suggested that the spin-1/2 three-leg spin tube has a 1/3 magnetization plateau. However, the magnetization measurement revealed that the compound [(CuCl 2 tachH) 3 Cl]Cl 2 has no magnetization plateau [3]. The density matrix renormalization group (DMRG) study [4] indicated that the spin-1/2 twisted three-leg spin tube has no plateau for sufficiently large exchange interaction along the chain.
In this paper, as another reason of the gapless structure, we consider the easy-plane anisotropy which stabilizes the 120-degree order. The 1/3 magnetization plateau is expected to vanish for sufficiently large easy-plane anisotropy. Thus we investigate the spin-1/2 twisted three-leg spin tube with the easy-plane anisotropy, using numerical diagonalization based on Lanczös algorithm.
We have successfully observed cases of the existence and the absence of the magnetization plateau at 1/3 height of the saturation magnetization when the anisotropy is tuned. And the phenomenological-renormalization analysis has clarified a quantum phase transition between the magnetization plateau and plateauless phases at this height.
The phase diagram is presented in Fig.5.

Model
We consider the spin-1/2 twisted three-leg spin tube, shown in Fig.1. This is a good theoretical model of the cluster compound [(CuCl 2 tachH) 3 Cl]Cl 2 [3] . The Hamiltonian is written as where S i,j is the spin-1/2 operator, 1 ( 2 ) denotes the intra-(inter-) triangle coupling which is supposed to be antiferromagnetic, and i (j) represents the label of the rung (leg) direction. Here i = 4 (i = 0) is regarded the same as i = 1 (i = 3). The XXZ anisotropy of the interactions is represented by λ. It is clear that the unit cell is composed of three spins from Fig.1(b). In this paper, we fix 1 to unity and consider the easy-plane anisotropy λ < 1) only.
The unit cell consists of three sites as shown in Fig.2. According to Oshikawa-Yamanaka-Affleck theorem, the necessary condition for the existence of magnetization plateau is the form -= integer (2) where S is the total spin quantum number and m is the magnetization per unit cell. Thus the three-leg spin tube possibly has the 1/3 magnetization plateau even without spontaneous translational symmetry breakdowns. The expected mechanism for the 1/3 plateau is shown in Fig.2(a). On the other hand, the easy-plane anisotropy tends to stabilize the 120°structure like Fig.2(b) [7]. Therefore, a quantum phase transition of the Berezinskii-Kosterlitz-Thouless (BKT) type between the plateau and gapless phases is expected to occur when λ decreases [8,9].
In this paper, we fixed 1 = 1.0 and changed 2 (0.2,…,0.5). We performed the numerical diagonalization based on the lanczös algorithm to calculate the lowest energy eigenvalue in each subspace characterized by = ∑ (M = 0,…,3L/2), which denoted as 0 ( , ). Using the calculated eigenvalues, the quantum phase transition is investigated. 120-degree structure expected to be realized for a strong easy-plane anisotropy.
The scaled gaps for L = 6 and 8 cross to each other and we can obtain (7), but the ones for L = 4 and 6 do not cross due to the finite-size effect. Thus we use (7) as the estimated phase boundary between the plateau and plateauless phases at 1/3 of the saturation magnetization. And we performed the same method to obtain (7) for various values of 2 .

Phase Diagram
The boundary between the 1/3 plateau and plateauless phases is shown in Fig.4. Also we can consider that the behavior of scaled gaps for L = 4 and 6 indicates the existence of the plateau in the whole region due to a serious finite size effect. Since the phase boundary was obtained only for one system size (L = 7), we could not estimate the boundary in the thermodynamic limit. The caluculation for larger systems would be desirable. However, even within the present analysis, we can expect the region of the plateauless phase should be larger in the thermodynamic limit because the finite-size effect overestimates the plateau phase. Therefore we can consider that the present estimation of the phase boundary would give a lower limit of the correct one in the phase diagram. Fig.4 The phase diagram in the λ-2 plane. Since the finite size effect tends to overestimate the plateau phase, the present result is expected to give a lower limit of the phase boundary in the thermodynamic limit.

Discussion and Summary
We have investigated the spin-1/2 twisted three-leg antiferromagnetic quantum spin tube in the presence of the easy-plane anisotropy, using the numerical diagonalization of finite-size clusters and the PRG analysis. It has been pointed out that the PRG analysis In summary, we have shown that the magnetization plateau of the spin-1/2 twisted three-leg spin tube vanishes by the introduction of the easy-plane anisotropy by estimating the lower limit boundary of the plateau region for the first time. More accurate determination of the phase diagram is left for future works.