EFFECTIVE HAMILTONIAN IN THE PROBLEM OF A CENTRAL SPIN-COUPLED TO A SPIN ENVIRONMENT

We consider here the problem of a ''giant spin'', with spin quantum nu mber S much greater than 1, interacting with a set of microscopic spin s. Interactions between the microscopic spins are ignored. This model describes the low-energy properties of magnetic grains or magnetic mac romolecules (ferromagnetically or antiferromagnetically ordered) inter acting with a surrounding spin environment, such as nuclear spins. Our aim is to give a general method for truncating the model to another o ne, valid at low energies, in which a two-level system interacts with the environmental spins, and higher energy terms are absorbed into a n ew set of couplings. This is done using an instanton technique. We the n study the accuracy of this technique, by comparing the results for t he low energy effective Hamiltonian, with results derived for the orig inal giant spin, coupled to a macroscopic spin, using exact diagonaliz ation techniques. We find that the low energy central spin effective H amiltonian gives very accurate results (with increasing accuracy for l arge S), provided the typical coupling energies between the giant spin and the microscopic spins are not too large, and provided temperature and external field are sufficiently low. The essential limitation to the applicability of the low-energy effective Hamiltonian is just the semiclassical WKB approximation itself, which inevitably fails for ver y small S. Our results thus justify previous use of this effective Ham iltonian in calculations of the effects of nuclear spins on the dynami cs of nanomagnetic systems.