Spin Glasses: Recent Theoretical Developments

Abstract

The considerable recent interest in phase transitions in random magnetic systems is reflected in there being several lectures on this topic presented here (Aharony, Coles, Cowley and myself). From a theoretical point of view, probably the simplest model which can describe the properties of most of these systems has the Hamiltonian

$$H = \mathop{{ - \sum }}\limits_{{ < ij > }} {{J}_{{ij}}}{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{S}}}_{i}} \cdot {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{S}}}_{j}} - \mathop{\sum }\limits_{i} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h}}}_{i}} \cdot {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{S}}}_{i}}$$

(1)

where ${{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{S}}}_{i}}$ is an m-component classical spin on site i of a d-dimensional lattice, J_ij is a pair interaction and ${{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h}}}_{i}}$ is a local field. If ${{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h}}}_{i}}$ is independent of site i (a uniform field) and J_ij are random, the model can represent a system with two types of magnetic atoms A and B. If one species in non-magnetic then ordering disappears when the concentration of this species exceeds the percolation concentration (discussed by Aharony). This type of behaviour is fairly well understood. Another very relevant case is where the J_ij are not random but the local fields are disordered, and on average, point equally in all directions. Aharony and Cowley will discuss this more controversial topic. Spin glasses are characterised by extreme disorder in the interactions J_ij such that both signs occur with significant probability and in a random way. Professor Coles will describe systems in nature which have this property.