Abstract: Mean-field models of 2-spin Ising spin glasses with interaction matrices taken from ensembles which are invariant under $O(N)$ transformations are studied \cite{cdl1,cdl2}. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the $O(N)$ disorder and one thus finds another way of doing the Itzykson-Zuber integral over $O(N)$ \cite{itzu}. The analytic results are confirmed by extensive Monte Carlo simulations for large system sizes and by exact enumeration for small system sizes.

We analyze a class of mean field spin glass models with Hamiltonian
 

H = -\frac{1}{2}\sum_{ij} J_{ij} S_i S_j

where the $S_i$ are $N$ Ising spins. The interaction matrix $J$ is constructed via the following procedure
 

 J = {\cal O}^T\Lambda {\cal O}
 

where ${\cal O}$ is a random $O(N)$ matrix chosen with the Haar measure. The matrix $\Lambda$ is diagonal with elements independently chosen from a distribution $\rho(\lambda)$. The support of $\rho(\lambda)$ is taken to be finite and independent of $N$, this ensures the existence of the thermodynamic limit. These models are generalised forms of the random orthogonal models introduced by Marinari, Parisi and Ritort \cite{mapari}. The interest of this kind of model is that one may average over the $O(N)$ disorder ${\cal O}$ and then examine the nature of the spin glass phase as a function of the eigenvalue distribution $\rho(\lambda)$. In particular we show that the way in which $\rho(\lambda)$ vanishes at the maximal value of its support, $\lambda_{\max}$, determines whether the glass transition is a classical spin glass transition at a temperature $T_c$ or a structural glass transition. More precisely, if the density of states of $\Lambda$ $\rho(\lambda)\sim (\lambda_{\max} -\lambda)^\gamma$ near $\lambda_{max}$ then $T_c = 0$ for $\gamma \leq 0$ but a finite temperature second order phase transition is possible for $\gamma > 0$. This finite temperature classical spin glass transition only occurs if the same model but with spherically constrained spins (such that $S_i \in (-\infty,\infty)$ and $\sum_i S_i^2 = N$) exhibits a finite temperature phase transition. Where this is not the case we study the system using a one step replica symmetry breaking scheme to determine the dynamical transition temperature $T_D$ and the Kauzmann temperature $T_K$. Numerical simulations are carried out to confirm our analytical predictions on this class of models. We carry out both Monte Carlo simulations and exact enumeration calculations. The dynamical transition temperature $T_D$ estimated from simulations agrees well with our analytic calculations. The exact enumeration carried out on small system sizes confirms the dynamical nature of the transition occurring at $T_D$.