New phenomena in the random field Ising model

We reconsider Ising spins in a Gaussian random field within the replica formalism. The corresponding continuum model involves several coupling constants beyond the single one which was considered in the standard $\phi^4$ theory approach. These terms involve more than one replica, and therefore in a mean-field theory they do not contribute to the zero-replica limit. However, the fluctuations involving those extra terms are singular on the Curie line below eight dimensions, and by the time one reaches the dimension six, it is necessary to keep them in the renormalization group analysis. As a result it is found that there is no stable fixed point of order $(6-d)$. Whether this means that there is no expansion in powers of $(6-d)$, or that the transition is driven to first order by these fluctuations, is difficult to decide at this level, but it explains the failure of the $(d, d-2)$ correspondence