Ferromagnetic Phase Transitions in Random Fields: The Breakdown of Scaling Laws

The scaling laws relating the critical exponents characterizing the ferromagnetic phase transition are shown to be modified in the presence of quenched random magnetic fields. The hyperscaling relation $d\nu =2-\alpha$, for example, becomes $(d+{\lambda }_u)\nu =2-\alpha$, where the index ${\lambda }_u$ is negative and is related to the range of the ferromagnetic exchange interactions. This breakdown of hyperscaling results from a singular dependence of the thermodynamic functions on the scaling field associated with an irrelevant operator.