O ct 2 01 9 Magnetization Plateau of the Distorted Diamond Spin Chain

Abstract The frustrated quantum spin system on the distorted diamond chain lattice is investigated using the numerical diagonalization of finite-size clusters and the level spectroscopy analysis. In the previous work this system was revealed to exhibit the 1/3 magnetization plateau due to two different mechanisms depending on the coupling parameters, and the phase diagram at the 1/3 magnetization was obtained. In the present work it is found that the 1/3 magnetization plateau vanishes for sufficiently large XY -like coupling anisotropy. The phase diagram based on the level spectroscopy analysis is also presented.


I. INTRODUCTION
The magnetization plateau is one of interesting phenomena in the field of the strongly correlated electron systems. It possibly appears as the quantization of magnetization, when the one-dimensional quantum spin system satisfies the necessary condition where S is the total spin and m is the magnetization per unit cell 1 . The S=1/2 distorted diamond spin chain 2 is a strongly frustrated quantum spin system which exhibits the 1/3 magnetization plateau according to the condition. This system was proposed as a good theoretical model of the compound Cu 3 (CO 3 ) 2 (OH) 2 , called azurite 3 . Actually the magnetization measurement of the azurite detected a clear magnetization plateau at 1/3 of the saturation magnetization. The theoretical study by the numerical exact diagonalization 4 suggested that the 1/3 magnetization plateau is induced by two different mechanisms. One is based on the ferromagnetic mechanism and the other is due to the formation of singlet dimers and free spins. In the case of the azurite the 1/3 plateau is supposed to be due to the latter mechanism. In some previous theoretical works 5-8 on quantum spin systems it was reported that the magnetization plateau disappears in the presence of sufficiently large XY -like (easy-plane) coupling anisotropy. Thus in this paper we introduce the XY -like anisotropy to the S = 1/2 distorted diamond spin chain and consider the stability of the 1/3 magnetization plateau against the anisotropy in the case of both mechanisms. We note that the anisotropy inversion phenomena was found in the distorted diamond chain with the XXZ anisotropy 9,10 . For this purpose we use the numerical exact diagonalization of finite-size clusters and the level spectroscopy analysis to find the quantum phase transition between the plateau and the no-plateau phases.

We investigate the model described by the Hamiltonian
where S j is the spin-1/2 operator, J 1 , J 2 , J 3 are the coupling constants of the exchange interactions, and λ is the coupling anisotropy. The schematic picture of the model is shown in Fig. 1. In this paper we consider the case of the XY -like (easy-plane) anisotropy, namely

IV. LEVEL SPECTROSCOPY ANALYSIS AND PHASE DIAGRAM
In order to detect the quantum phase transitions among the plateaux-A, -B and noplateau phases, the level spectroscopy analysis 11,12 is one of the best methods. According to this analysis, we should compare the following three energy gaps; The level spectroscopy method indicates that the smallest gap among these three gaps for  Fig. 2 for L =4, 6 and 8.
Assuming that the finite-size correction is proportional to 1/L 2 , we estimate the phase boundaries in the thermodynamic limit from every level-cross point. The phase diagram in the J 2 -λ plane at m = 1/3 for J 1 = 1.0 and J 3 = 0.1 is shown in Fig. 3.
It suggests that both plateaux are so stable against the XY -like anisotropy that the plateaux vanish at quite large negative λ.

V. CONCLUDING REMARKS
Using the numerical exact diagonalization and the level spectroscopy analysis, the S = 1/2 distorted diamond spin chain is investigated. It is found that the 1/3 magnetization plateau vanishes for sufficiently large negative XY -like anisotropy, in both cases of the ferrimagnet-like and the dimer-monomer plateaux. A typical phase diagram at m = 1/3 is presented in Fig.3.
One of the remarkable natures of the plateau phase diagram Fig.3 is the survival of the 1/3 plateau to the λ < 0 region, where the direct S z − S z interaction is ferromagnetic.
The essential mechanism of the plateau-A is the Lieb-Mattis ferrimagnetism 13 . In the simple S = 1/2 XXZ chain with λ < 0, although the direct S z −S z interaction is ferromagnetic, the antiferromagnetic S z − S z correlation survives to λ = −1 induced by the antiferromagnetic We think that the no-plateau state is essentially the Tomonaga-Luttinger liquid state in the magnetic field. We can see a sudden decrease of the phase boundary between the plateau-A and the no-plateau J 2 = 0.2. The physical explanation for this phenomena is a future problem. We believe that our plateau phase diagram is important for the full understanding of the distorted diamond chain and will yields important informations if the distorted diamond chain with the XXZ anisotropy is found or synthesized in future.