Renormalization group theory of crossovers

Crossover phenomena are both ubiquitous and important. In this report, we review the foundations of renormalization group theory as applied to crossover behavior and consider several paradigmatic applications. We confine ourselves to situations where the crossover can be described in terms of an effective field theory, in particular concentrating on the prototypical example of an O(N) model in a constrained geometry or at finite temperature. Calculation of universal crossover scaling functions is considered where we show how the renormalization group can in principle be applied to the latter to obtain expressions as accurate as those of standard universal quantities, such as critical exponents and amplitude ratios. Particular attention is paid to the scaling equation of state for an O(N) model.