Disordered Systems

The properties of disordered systems are much more difficult to unravel than those of regular (e.g. crystalline) systems that possess a degree of translational invariance. The properties of such systems are often unexpected. As long ago as 1958, P. W. Anderson showed that the wavefunctions of electronic states in disordered systems can be extended or localised depending on the energy of the state. More recently, critical phenomena in disordered systems (e.g. disordered ferromagnets) have been discovered to have some "non-self-averaging" properties, i.e. properties that vary from sample to sample according to the particular realisation of the disorder.

A particularly challenging example of a disordered system is the "spin glass", whose properties are still controversial after some 30 years of study. A spin glass is a system containing localised magnetic moments, the pairwise interactions between moments varying from pair to pair according to their location in the system. Crucially, the interactions can be either ferromagnetic or antiferromagnetic and the competition arising from this leads to a new state - the "spin glass" - which is dynamically frozen but contains no conventional long-range order.

There are two main competing theories of the spin-glass state. One, due mainly to Parisi, is based on extending to real systems some results derived from a mean-field model, the Sherrington-Kirkpatrick model, which has many exotic properties including a large number of pure states, unrelated by symmetry, in the low-temperature phase. With M. A. Moore, I have been one of the instigators (along with W. McMillan, D. S. Fisher and D. A. Huse) of an alternative theory, the "droplet/scaling theory", based on the properties of low free-energy excitations in the system, which works on the basis of there being only two pure states. One of the cleanest theoretical descriminators between the theories is the effect of an applied magnetic field. The mean-field theory predicts that a transition to the spin-glass persists in the presence of a weak applied field, whereas in the droplet/scaling theory the transition is removed by the field (much as in a ferromagnet). Recent numerical studies, mainly by A. P. Young and coworkers, provides good evidence that there there is no transition, at least in systems of sufficiently low spatial dimensionality (d=3 and d=4). It is still an open question what happens in higher dimensions. An old renormalization group calculation of mine with S. A. Roberts suggests that a magnetic field removes the transition for d<6 but not for d>6.