A pure quantum mechanical theory of parity effect in tunneling and evolution of spins

In recent years, the spin parity effect in magnetic macroscopic quantum tunneling has attracted extensive attention. Using the spin coherent-state path-integral method it is shown that if the HamiltonianH of a single-spin system hasM - fold rotational symmetry around z-axis, the tunneling amplitude 〈−S|eHt|S〉 vanishes when S, the quantum number of spin, is not an integer multiple ofM/2, where |m〉 (m=-S, -S +1, ⋯, S) are the eigenstates of Sz. Not only is a pure quantum mechanical approach adopted to the above result, but also is extended to more general cases where the quantum system consists ofN spins, the quantum numbers of which can take any values, including the single-spin system, ferromagnetic particle and antiferromagnetic particle as particular instances, and where the states involved are not limited to the extreme ones. The extended spin parity effect is that if the Hamiltonian ℋ of the system ofN spins also has the above symmetry, then 〈m′N⋯m′2m′1|e−Ht|m1m2⋯mN vanishes when ∑i=1N (mi−m′1) not an integer multiple ofM, where |m1m2⋯mN〉=∏α=1N|ma〉 are the eigenstates of Saz. In addition, it is argued that for large spin the above result, the so-called spin parity effect, does not mean the quenching of spin tunneling from the direction of ⊕-z to that of ±z.