Computer studies of local minima of the planar spin glass

Abstract

The nature of the local minima of the 'planar' spin glass Hamiltonian at zero temperature in two and three dimensions is investigated by means of computer simulation. The authors find that the eigenvalue density rho ( lambda ) of the Hessian varies as lambda1/2for small lambda in three dimensions but that only approximately 40% of this can be accounted for by a spin-wave argument. In two dimensions all the eigenvectors appear to be localised but in three dimensions there appears to be a sharp transition from extended to localised states at a critical value of lambda . The addition of a uniform magnetic field or uniaxial or cubic anisotropy also produces in three dimensions localised states at small values of lambda , together with a large reduction in rho ( lambda ) as lambda to 0. The authors deduce that the 'hole' picture of the dynamics (based on the assumption that the system stays in the vicinity of a particular local minimum for macroscopic periods) is not likely to be an accurate description at long times.