Critical behavior of randomn≥4vector models studied by the renormalization-group technique in4−εdimensions: Crossover from first-order to smeared transitions

The phase transitions in a large class of physical systems are described by $n\ge 4$ component order parameters. Here, the critical behavior of quenched random $n\ge 4$ vector models is studied by means of renormalization-group theory in $4-\epsilon$ dimensions. Recursion relations for average potentials are constructed following the methods derived by Lubensky. For several Hamiltonians describing homogeneous $n\ge 4$ systems there exist no stable fixed points in $4-\epsilon$ dimensions which explains the first-order transitions actually observed in these systems. It is shown that the recursion relations for the corresponding quenched random systems are also unstable. However, the runaway in this case is of a fundamentally different nature. The fluctuations of the local mean-field transition temperature diverge, and this behavior is interpreted as a "smeared" transition. This interpretation is consistent with existing experiments. On the other hand, in the cases where the homogeneous Hamiltonian possesses a stable fixed point in $4-\epsilon$ dimensions, this fixed point remains stable against random perturbations, so no change in the critical behavior is expected. For most of the models studied there is at least one fixed point of order ${\epsilon }^{\frac{1}{2}}$. These fixed points are all unstable. It is suggested that experiments should be performed to determine the critical behavior of random $n\ge 4$ systems. Of particular interest are the systems with no stable fixed points, such as Cr, Eu, MnO, and ${\mathrm{UO}}_2$, where crossover from first-order to a smeared transition is predicted.