Studies in random Ising spin systems

We give a brief survey of Ising spin systems in the presence of random bonds or random fields. Specific calculations are done on the twodimensional short-ranged random-bond model and the three-dimensional random-field model. The phenomenological renormalization group and the transfer-matrix method are generalized to random systems and applied to the random-bond Ising model, also called the Edwards-Anderson spin-glass model. Results show that the two-dimensional Edwards-Anderson model has a spin-glass transition at zero temperature, in agreement with other calculations. We obtain an accurate estimate of the correlation-length exponent. We also find unexpectedly different critical behavior for different bond distributions. A simple renormalization-group calculation is applied to the twodimensional random-bond Ising model with non-zero-mean bond strength. Our result fails to confirm McMillan's result that there are two different critical behaviors along the ferromagnetic phase boundary. McMillan's domain-wall renormalization group is applied to the three -dimensional random-field Ising model. Numerical results show that the critical behavior of this model is controlled by a fixed point at zero temperature. The exponents are estimated. We find that, whenever there is a relevant zero-temperature fixed point, the singular part of the free energy is not of the usual form: The leading exponent that describes the renormalization-group flow towards zero temperature is relevant to the critical behavior. One of the consequences is that the hyperscaling law is modified. This new picture explains many experimental results for dilute antiferromagnets, ·which are experimental realizations of the random-field Ising model. We also discuss the possibility of a first-order transition for this model.U of I OnlyThesi