Absence of a metallic phase in random-bond Ising models in two dimensions:Applications to disordered superconductors and paired quantum Hall states - art. no. 024404

When the two-dimensional random-bond Ising model is represented as a nonint eracting fermion problem, it has the same symmetries as an ensemble of rand om matrices known as class D. A nonlinear sigma model analysis of the latte r in two dimensions has previously led to the prediction of a metallic phas e, in which the fermion eigenstates at zero energy are extended. In this pa per we argue that such behavior cannot occur in the random-bond Ising model , by showing that the Ising spin correlations in the metallic phase violate the bound on such correlations that results from the reality of the Ising couplings. Some types of disorder in spinless or spin-polarized p-wave supe rconductors and paired fractional quantum Hall states allow a mapping onto an Ising model with real but correlated bonds, and hence a metallic phase i s not possible there either. It is further argued that vortex disorder, whi ch is generic in the fractional quantum Hall applications, destroys the ord ered or weak-pairing phase, in which non-Abelian statistics is obtained in the pure case.