The spin nematic order has attracted a lot of interest in the field of magnetism. It is a kind of multipole order of spins. The previous theoretical and numerical studies[1-3] predicted that the spin nematic order would be induced by the frustration of the ferromagnetic and antiferromagnetic exchange interactions, or the biquadratic interaction. The spin nematic order is characterized by the long-range four spin correlation and the two-magnon bound state. The previous numerical diagonalization and the finite-size scaling study had indicated that a two-magnon bound state can occur in the S=1 antiferromagnetic chain with the single-ion anisotropy under magnetic field[4].The recent analysis of the critical exponents of the spin correlation functions[5] suggested that this two-magnon bound state includes the spin nematic liquid phase, as well as the spin density wave (SDW) liquid one. The phase diagrams with respect to the anisotropy and the magnetization were obtained by the numerical diagonalization of finite size clusters. The same numerical analysis indicated that the spin nematic liquid phase appears in the magnetization process of the 1/2 spin ladder system with the anisotropic ferromagnetic rung interaction[6,7].These systems would be able to exhibit the spin nematic liquid even without the frustration or the biquadratic exchange interaction. In the present study, the S=1/2 distorted diamond spin chain is investigated using the numerical diagonalization of finite-size clusters. This system is a typical frustrated system. The recently discovered candidate material of this system, K3Cu3AlO2(SO4)4 called alumoklyuchevskite[8], includes the ferromagnetic interactions. Thus we investigated the distorted diamond chain with the ferromagnetic interactions, as well as the coupling anisotropy. As a result we found that a two-magnon bound state appears in the magnetization process. We also discus about the spin nematic liquid behaviors. 

**References** [1] A. F. Andreev and A. Grishchuk, Sov. Phys. JETP 60, 267 (1984). 
[2] H. H. Chen and P. M. Levy, Phys. Rev. Lett. 27, 1383 (1971). 
[3] T. Hikihara et al., Phys. Rev. B 78, 144404 (2008). 
[4] T. Sakai, Phys. Rev. B 58, 6268 (1998). 
[5] T. Sakai, H. Nakano, K. Okamoto and R. Furuchi, J. Phys.: Conf. Ser. 2164, 012030 (2022). 
[6] T. Sakai, K. Okamoto and T. Tonegawa, phys. Status solidi B 247, 583 (2010) 
[7] T. Sakai et al., in preparation. 
[8] M. Fujihara et al., Sci. Rep. 7, 16785 (2017).

Spin Nematic Liquid of the S=1/2 Distorted Diamond Spin Chain in Magnetic Field

The frustrated quantum spin systems exhibit various exotic magnetic phenomena. One of interesting phenomena is the magnetization plateau, which corresponds to the quantization of the magnetization. The S=1/2 distorted diamond spin chain is a typical one-dimensional frustrated systems. The previous theoretical study, using the numerical diagonalization of finite-size culsters and the level spectroscopy analysis[1], indicated the S=1/2 distorted diamond spin chain can exhibit the 1/3 magnetization plateau due to two different mechanisms, depending on the ratio of the exchange interactions. Azurite Cu3(OH)2(CO3)2 is a good candidate material of the distorted diamond spin chain. The high-field magnetization measurement successfully revealed that azurite exhibits the 1/3 magnetization plateau[2]. Recently as another good candidate material alumoklyuchevskite K3Cu3AlO2(SO4)4 was discovered[3]. In this compound ferromagnetic exchange interactions exist as well as antiferromagnetic ones, while azurite has only antiferromagnetic ones. On the other hand, if the anisotropic ferromagnetic interaction is introduced to the distorted diamond spin chain, another mechanism of the 1/3 magnetization plateau would be expected, based on the previous theoretical study on the mixed spin chain[4]. Thus it would be interesting to study the model of the distorted diamond spin chain with the anisotropic ferromagnetic interaction, to consider the magnetization plateau. In the present work, we investigated this model using the numerical diagonalization and the level spectroscopy analysis and obtained the phase diagram at 1/3 of the saturation magnetization, which includes the plateau phase due to a new mechanism for the distorted diamond spin chain.

Magnetization Plateau of the Distorted Diamond Spin Chain with Anisotropic Ferromagnetic Interactions

The spin nematic phase, which is a kind of multipole phases, has attracted a lot of interest in the field of the strongly correlated electron systems, as well as the quantum spin liquid phase. Using the numerical exact diagonalization, the density matrix renormalization group (DMRG) calculation, and the finite-size scaling analysis, it is found that some spin nematic and spin liquid phases are induced by external magnetic field in the anisotropic and frustrated quantum spin systems. In our previous work[1], it was found that a field-induced nematic phase appears at some critical field in the anisotropic spin ladder. The nematic phase is characterized by the power-law decay in the correlation function of the second-order spin moment. In addition at some higher critical field a quantum phase transition can occur to the conventional field-induced Tomonaga-Luttinger liquid. Several typical magnetization curves calculated by DMRG are presented. Recently the field-induced nematic phase was observed on the frustrated spin ladder system[2]. So we studied on a frustrated spin ladder system[3], using the numerical diagonalization and DMRG. As a result, it was found that several exotic quantum phases, including the spin-nematic liquid phase. We also reported several interesting phase diagrams of this model and some related systems[4,5]. In the present paper, we investigate the S=1 antiferromagnetic chain with the biqadratic interaction, as another candidate that exhibits the field-induced spin nematic liquid pahse. The numerical diagonalization study of finite-size clusters indicates that the spin nematic liquid pahse appears in this model under sufficiently strong magnetic field. The phase diagram in the plane of the magnetic field and the biquadratic interaction is presented.References: [1] T. Sakai, T. Tonegawa and K. Okamoto, Physica Status Solidi B 247, 583 (2010) . [2] N. Buttgen et al., Phys. Rev B 90, 134401 (2014) . [3] T. Hikihara, T. Tonegawa, K. Okamoto and T. Sakai, J. Phys. Soc. Jpn. 86, 054709 (2017). [4] T. Tonegawa, T. Hikihara, K. Okamoto, S. C. Furuya, and T. Sakai, J. Phys. So. Jpn. 87, 104002 (2018). [5] T. Sakai, K. Okamoto and T. Tonegawa, Phys. Rev. B 100, 054407 (2019) 

Quantum Spin Nematic Liquid in the S=1 Antiferromagnetic Chain with the Biquadratic Interaction

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Spin Nematic Liquid of the S=1/2 Distorted Diamond Spin Chain in Magnetic Field

Magnetization Plateau of the Distorted Diamond Spin Chain with Anisotropic Ferromagnetic Interactions

Quantum Spin Nematic Liquid in the S=1 Antiferromagnetic Chain with the Biquadratic Interaction

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