We study structure of the gapless points (diabolical points) at zero magnetic field $\left(H_z=0\right)$ of single-spin models with spin anisotropies. Nontrivial appearance of diabolical points at finite transverse field $H_x$ has been studied from the viewpoint of interference of the Berry phase and related phenomena have been experimentally found in the single molecular magnet ${\text{Fe}}_8$. We study effects of the orthorhombic single-ion anisotropy $E\left(S_{+}^2+S_{-}^2\right)$ 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 $C\left(S_{+}^4+S_{-}^4\right)$ and find that, in contrast to the semiclassical limit $\left(S\to \infty \right)$, location of a pair annihilation of the diabolical point does not coincide 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 $\left(C,H_x\right)$ plane.

• Received 4 October 2008

DOI:https://doi.org/10.1103/PhysRevB.78.214434