Abstract
Theoretical approaches to spin glasses can be divided into two main categories. The first approach envisages constructing the mean-field solution of (say) the Edwards-Anderson1(EA) Hamiltonian and then systematically expanding about it to describe the properties of three-dimensional spin glasses. Producing a mean-field theory is equivalent to solving the Sherrington-Kirkpatrick2 (SK) spin-glass model. This model is now well understood and the solution reveals a rich structure of many pure states related by an ultrametric topology3. Fig 1(a) shows the expected phase diagram in a field. The SK model is the infinite dimensional limit of the EA Hamiltonian. Recent studies by Kondor4 suggest that the ultrametric behaviour, de Almeida-Thouless (AT)5 line etc. will not exist below six dimensions. Thus, the program of expanding about the customary mean-field solution to obtain the properties of spin glasses whose dimensionality is less than six does not look promising.
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