Magnetic multipolar order, especially, quadrupolar (spin nematic) order, is one of the current topics in frustrated magnetism. Recently, frustrated spin-1/2 chains with ferromagnetic nearest neighbor J1(<0) and antiferromagnetic next nearest neighbor J2(>0) couplings have been theoretically shown to exhibit a spin nematic quasi long range order in a broad range of J1/J2 as external magnetic field is applied [1,2]. In addition, magnetic properties of several quasi one-dimensional (1D) cuprate magnets [3] are believed to be described by this J1-J2 spin chain model [1,2,4,5]. In particular, recent experiments show that LiCuVO4, one of the quasi-1D cuprates, possesses a new phase near saturation, and it is expected to be a 3D spin nematic ordered phase [6,7]. Motivated by these results, we have developed a general theory of the spin nematic ordering [8] for 3D spatially anisotropic magnets consisting of weakly coupled J1-J2 spin chains, combining the field theoretical technique [1,4] with numerical results. We have completed the phase diagram in a wide temperature and magnetic-field ranges. The phase diagram contains both spin nematic and spin-density-wave ordered phases, and these two orders compete with each other. We will discuss some universal features of the phase diagram and the relevance of our result to LiCuVO4.

[1]T.Hikihara, L.Kecke, T.Momoi, and A.Furusaki, PRB78, 144404 (2008).
[2]J.Sudan, A.Luscher, and A.M.Lauchli, PRB80, 140402(R) (2009).
[3]See, for example, M.Hase, et al, PRB70, 104426 (2004); M.Enderle, et al, EPL70, 237 (2005); Y.Yasui, et al, JPSJ80, 033707 (2011); T.Masuda, et al, JPSJ80, 113705 (2011).
[4]M.Sato, T.Momoi, and A.Furusaki, PRB79, 060406(R) (2009); M.Sato, T.Hikihara, and T.Momoi, PRB83, 064405 (2011).
[5]S.Furukawa, M.Sato, and S.Onoda, PRL105, 257205 (2010).
[6]L.E.Svistov, et al, JETP Lett. 93, 21 (2011).
[7]M. Mourigal, et al, PRL109, 027203 (2012).
[8]M.Sato, T.Hikihara, and T.Momoi, arXiv:1208.2235.