The equivalence between discrete-spin Hamiltonians and Ising Hamiltonians with multi-spin interactions

An argument is presented to explain the fact that the same mean-field results apply to several spin glass models, such as the 'random-energy' model and the large-p Potts glass model. The dilute simplest spin glass model is solved, yielding the same mean-field results. The argument is based on the fact that the symmetry of the discrete-spin system can be determined by the discrete-spin type or by the form of the interactions. The mappingbetween the models can be extended to all forms of discrete-spin systems. Magnetic systems with random competing interactions often condense at low temperature into a spin glass (SG) phase. The mean-field (MF) theory based on the EdwardsAnderson model [l] reveals that the transition of the low-temperture phase is a continuous transition, similar to an ordinary second-order phase transition. However, the SG phase is very unusual. It consists of many degenerate partially overlapping pure states characterised by an order parameter q ( x ) . The inverse of the order parameter, x ( q ) , determines the probability that two pure states have an overlap [2 ,3 ] less than or equal to q. Recently, there has been a growing tendency to apply concepts of SG theory to various infinite-range discrete SG models [4-91 which are a generalisation of the SK model [5] . One form of generalisation consists of a spin with more than two components. Another form of generalisation is the replacement of the random pair interactions by multi-spin interactions. Several of these generalised SK models [4-91 have a similar MF solution. In this Letter, we show why models that differ in the spin representation and in the form of the interactions among the spins nevertheless exhibit a similar behaviour in their MF solution. The argument is based on the fact that each discrete-spin Hamiltonian can be mapped into an equivalent Ising Hamiltonian with multi-spin interactions. In this mapping each discrete spin o f p discrete states is represented by a block of log2p Ising spins. We present now examples of systems that have a similar MF solution. The system of Np-state Potts variables a(i) = 1 , 2 , . . . , p is described by the Hamiltonian [4,5]