The metastable minima of the Heisenberg spin glass in a random magnetic field

We have studied zero temperature metastable states in classical $m$-vector component spin glasses in the presence of $m$-component random fields (of strength $h_{r}$) for a variety of models, including the Sherrington Kirkpatrick (SK) model, the Viana Bray (VB) model and the randomly diluted one-dimensional models with long-range power law interactions. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case and the quenched case, both for fields above and below the de Almeida Thouless (AT) field ($h_{AT} > 0$ for $m>2$). We have done quenches starting from a random initial state by putting spins parallel to their local fields until convergence and found that in zero field it always produces minima which have zero overlap with each other. For the $m=2$ and $m=3$ cases in the SK model the final energy reached in the quench is very close to the energy $E_c$ at which the overlap of the states would acquire replica symmetry breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states $\rho(\lambda)$ of the Hessian matrix in the annealed approximation. Despite the absence of continuous symmetries, the spectrum extends down to zero with the usual $\sqrt{\lambda}$ form for the density of states for $h_{r}<h_{AT}$. However, when $h_{r}>h_{AT}$, there is a gap in the spectrum which closes up as $h_{AT}$ is approached. For the VB model and the other models our numerical work shows that there always exist some low-lying eigenvalues and there never seems to be a gap. There is no sign of the AT transition in the quenched states reached from infinite temperature for any model but the SK model, which is the only model which has zero complexity above $h_{AT}$.Comment: 16 pages, 8 figures (with modifications), rewritten text and abstrac

Origin of the Growing Length Scale in M-p-Spin Glass Models

Two versions of the M-p-spin glass model have been studied with the Migdal-Kadanoff renormalization group approximation. The model with p=3 and M=3 has at mean-field level the ideal glass transition at the Kauzmann temperature and at lower temperatures still the Gardner transition to a state like that of an Ising spin glass in a field. The model with p=3 and M=2 has only the Gardner transition. In the dimensions studied, d=2,3 and 4, both models behave almost identically, indicating that the growing correlation length as the temperature is reduced in these models -- the analogue of the point-to-set length scale -- is not due to the mechanism postulated in the random first order transition theory of glasses, but is more like that expected on the analogy of glasses to the Ising spin glass in a field.Comment: 5 pages, 3 figures, revised versio