On the eigenvalue spectrum of the susceptibility matrix for random spin systems

The eigenvalues and eigenvectors of the inverse of the susceptibility matrix are discussed for random spin systems. At a phase transition precipitated by the vanishing of the smallest eigenvalue it is shown that the corresponding eigenvector must be extended, and that the density of states must vanish at zero eigenvalue. Above the transition temperature, generalised Griffiths singularities are associated with the existence of localised states with arbitrarily small eigenvalues. For infinite spin dimensionality the eigenvalue problem is equivalent to an Anderson problem with correlated diagonal and off-diagonal disorder.