New phenomena in the random field Ising model

TL;DR

This paper investigates the critical behavior of the random field Ising model using the replica formalism, revealing that additional coupling terms become significant below six dimensions and challenge the existence of a stable fixed point.

Contribution

It introduces a comprehensive renormalization group analysis considering multiple couplings in the continuum model, highlighting the failure of the $(6-d)$ expansion and implications for phase transition nature.

Findings

No stable fixed point of order (6-d) found.

Additional replica couplings become relevant below six dimensions.

Failure of the $(d, d-2)$ correspondence explained.

Abstract

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. In this work we have not considered the possibility of a glassy phase in the would-be paramagnetic region, but in a subsequent work we shall present evidence for replica symmetry breaking in dimension lower than six.