A generalized Gross-Pitaevskii equation adapted to the $U(5)supset SO(5)supset SO(3)$ symmetry has been derived and solved for the spin-2 condensates. The spin-textile and the degeneracy of the ground state (g.s.) together with the factors affecting
the stability of the g.s., such as the gap and the level density in the neighborhood of the g.s., have been studied. Based on a rigorous treatment of the spin-degrees of freedom, the spin-textiles can be understood in a $N$-body language. In addition to the ferro-, polar, and cyclic phases, the g,s, might in a mixture of them when $0< M< 2N$ ($M$ is the total magnetization). The great difference in the stability and degeneracy of the g.s. caused by varying $varphi $ (which marks the features of the interaction) and $M$ is notable. Since the root mean square radius $R_{rms}$ is an observable, efforts have been made to derive a set of formulae to relate $R_{rms}$ and $% N$, $omega $(frequency of the trap), and $varphi $. These formulae provide a way to check the theories with experimental data.
The very-low-temperature thermal conductivity (kappa) is studied for BaCo_2V_2O_8, a quasi-one-dimensional Ising-like antiferromagnet exhibiting an unusual magnetic-field-induced order-to-disorder transition. The nearly isotropic transport in the lon
gitudinal field indicates that the magnetic excitations scatter phonons rather than conduct heat. The field dependence of kappa shows a sudden drop at sim 4 T, where the system unndergoes the transition from the Neel order to the incommensurate state. Another dip at lower field of sim 3 T indicates an unknown magnetic transition, which is likely due to the spin-flop transition. Moreover, the kappa(H) in the transverse field shows a very deep valley-like feature, which moves slightly to higher field and becomes sharper upon lowering the temperature. This indicates a magnetic transition induced by the transverse field, which however is not predicted by the present theories for this low-dimensional spin system.
In order to evaluate $g_0$, the interaction strength of a pair of $^{52}$Cr atoms with total spin S=0, a specially designed s-wave scattering of the pair has been studied theoretically. Both the incident atom and the target atom trapped by a harmonic
potential are polarized previously but in reverse directions. Due to spin-flip, the outgoing atom may have spin component $mu$ ranging from -3 to 3. The outgoing channels are classified by $mu$. The effect of $g_{0}$ on the scattering amplitudes of each of these $mu-$channels has been predicted.
