Topological quenching of spin tunneling in magnetic molecules with a fourfold easy axis

Spin tunneling is investigated in magnetic molecules that have an easy axis with fourfold symmetry, such as Mn₁₂-acetate, with emphasis on understanding the topological quenching of tunneling and the diabolical point locations in the magnetic field space. This is done using a model spin Hamiltonian that has a fourth-order term describing the tetragonal anisotropy. The problem is studied qualitatively using instantons and quantitatively using two methods: a discrete phase integral or Wentzel-Kramers-Brillouin method and perturbation theory in the fourth-order anisotropy and transverse magnetic field. The former method is used to find the splitting between various levels when the applied magnetic field is along the hard axis and is found to give good quantitative answers. The latter method is employed for fields which may have an easy component in addition to a hard one and is found to be effective in locating all the diabolical points. These points are found as the roots of a small number of polynomials in the hard component of the magnetic field and the basal plane anisotropy. These roots are used to obtain approximate formulas that apply to any system with total spin S ≤10. The analytic results are found to compare reasonably well with exact numerical diagonalization for the case of Mn₁₂-acetate. In addition, perturbation theory shows that the diabolical points may be indexed by the magnetic quantum numbers of the levels involved, even at large transverse fields. Certain points of degeneracy are found to be mergers (or near mergers) of two or three diabolical points because of the symmetry of the problem.