S=12XYmodel on cubic lattices. II. Longitudinal susceptibilities

The longitudinal susceptibility for the $S=\frac{1}{2}\mathrm{XY}$ model on cubic lattices is discussed on the basis of its relationship to the free energy. A series expansion for the susceptibility is developed and its behavior is studied at an asymptotic limit. The asymptotic behavior of the series expansion shows that the longitudinal susceptibility is nondivergent at the critical point $T_c$, its "singular" part behaving as ${(T-T_c)}^{-\alpha +1\mathrm{}}$, where $\alpha$ is the specific-heat exponent. A related quantity, the partial longitudinal susceptibility, which is important in the study of spin dynamics, is also shown to have a similar nondivergent critical behavior.