Multipolar phase in frustrated spin-1/2 and 1 chains

The $J_1-J_2$ spin chain model with nearest neighbor $J_1$ and next nearest neighbor anti-ferromagnetic $J_2$ interaction is one of the most popular frustrated magnetic models. This model system has been extensively studied theoretically and applied to explain the magnetic properties of the real low-dimensional materials. However, existence of different phases for the $J_1-J_2$ model in an axial magnetic field $h$ is either not understood or has been controversial. In this paper we show the existence of higher order $p>4$ multipolar phase near the critical point $(J_2/J_1)_c=-0.25$. The criterion to detect the quadrupolar or spin nematic (SN)/spin density wave of type two (SDW$_2$) phase using the inelastic neutron scattering (INS) experiment data is also discussed, and INS data of LiCuVO$_4$ compound is modelled. We discuss the dimerized and degenerate ground state in the quadrupolar phase. The major contribution of binding energy in the spin-1/2 system comes from the longitudinal component of the nearest neighbor bonds. We also study spin nematic/SDW$_2$ phase in spin-1 system in large $J_2/J_1$ limit.