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

Self-organized critical behavior and marginality in Ising spin glasses

We have studied numerically the states reached in a quench from various temperatures in the one-dimensional fully-connected Kotliar, Anderson and Stein Ising spin glass model. This is a model where there are long-range interactions between the spins which falls off a\ s a power $\sigma$ of their separation. We have made a detailed study in particular of the energies of the states reached in a quench from infinite temperature and their overlaps, including the spin glass susceptibility. In the regime where $\sigma \le 1/2$, where th\ e model is similar to the Sherrington-Kirkpatrick model, we find that the spin glass susceptibility diverges logarithmically with increasing $N$, the number of spins in the system, whereas for $\sigma> 1/2$ it remains finite. We attribute the behavior for $\sigma \le 1/\ 2$ to \emph {self-organized critical behavior}, where the system after the quench is close to the transition between states which have trivial overlaps and those with the non-trivial overlaps associated with replica symmetry breaking. We have also found by studying the d\ istribution of local fields that the states reached in the quench have marginal stability but only when $\sigma \le 1/2$.Comment: 16 pages, 7 figures. Modified format, new figures and a new section on marginality. Final published versio