Structure of energy level degeneracy of a single-spin model from a view point of symmetry of the spin anisotropy and its nontrivial spin(S)-dependence on the higher order anisotropy

We study structure of the gapless points (diabolical points) at zero magnetic field ($H_z=0$) of single-spin models with spin anisotropies. Nontrivial appearance of diabolical points at finite transverse field $H_x$ has been studied from the view point of interference of the Berry phase, and related phenomena have been experimentally found in the single molecular magnet Fe$_8$. We study effects of the orthorhombic single-ion anisotropy $E(S_+^2+S_-^2)$ and find a symmetry associated with the degeneracy, which provides a clear picture of the global structure of energy level diagram including the excited states. Moreover, we study effects of the higher order anisotropy $C(S_+^4+S_-^4)$, and find that, in contrast to the semiclassical limit $(S\to\infty)$, location of a pair annihilation of the diabolical point does not coincides with a point at which a pair of diabolical points appears in nonzero $H_y$ space(bifurcation points). Distance between the annihilation and bifurcation points vanishes when $S\to\infty$, which restores the semiclassical result. We obtain a complete structure of the diabolical points in the $(C,H_x)$ plane.