The applicability of mean field models of ferroelectric and ferromagnetic quantum critical points is examined for a selection of d-electron systems. Crucially, we find that the tendency of the effective interaction between critical fluctuation modes to become attractive and anomalous as the ordering temperatures tend to absolute zero results in particularly complex and striking phenomena. The multiplicity of quantum critical fields at the border of metallic ferromagnetism, in particular, is discussed here. 1 Introduction We consider the nature of quantum phase transitions driven by changes in composition of materials or changes in applied pressure, magnetic field or electric field in the low temperature limit. Quantum phase transitions exhibit surprisingly subtle and complex behaviour even in comparatively simple examples of cubic ferroelectric materials and ferromagnetic metals of high purity, which will be the main focus of this article.Early descriptions of quantum critical points (QCP), developed independently in ferroelectric materials [1] and ferromagnetic metals [2][3][4][5][6] in the 1970s, were based essentially on f 4 -quantum field models. They differ from the Ginzburg-Landau-Wilson models of classical critical phenomena by the inclusion of the dynamics of the order parameter field f(r,t), which represents a coarse-grained electric or magnetic polarization as a function of spatial coordinate r and temporal coordinate t (the imaginary time, which has a finite range at non-zero temperatures, 0 < t < h=k B T) [7]. The inclusion of the thermal coordinate increases the relevant dimension from the spatial dimension d to the effective dimension d eff ¼ d þ z, where z is the dynamical exponent defining the dispersion relation, i.e., the wavevector dependence of the frequency spectrum of fluctuations of the field f at small wavevectors (see x2).The self-consistent-field approximation, which applies in the case of classical critical phenomena for d > 4 in the classical f 4 model, might apply under a less restrictive condition d > 4 -z in the f 4 quantum treatment of critical