Quantum mechanical tunneling in the Heisenberg model with weak anisotropy

It is shown that the energy levels of low lying states in the Heisenberg model with weak anisotropy can be found as eigenvalues of a paramagnetic Hamiltonian whose anisotropy constants have been renormalized relative to the original ones and whose spin is proportional to the number of particles. The spectrum of such a system is rigorously described with the help of a simple effective potential permitting the use of familiar quantum-mechanical methods. Using instanton techniques, the splitting of the ground state energy due to tunneling is calculated and found to be quite sensitive to all parameters in the problem except the isotropic exchange constant