We investigate the controversial issue of the existence of universality classes describing critical phenomena in three-dimensional systems characterized by a matrix order parameter with symmetry O(2)⊗O(N ) and symmetry-breaking pattern O(2)⊗O(N )→O(2)⊗O(N − 2). Physical realizations of these systems are, for example, frustrated spin models with noncollinear order.Starting from the field-theoretical Landau-Ginzburg-Wilson Hamiltonian, we consider the massless critical theory and the minimal-subtraction scheme without ǫ expansion. The three-dimensional analysis of the corresponding five-loop series shows the existence of a stable fixed point for N = 2 and N = 3, confirming recent field-theoretical results based on a six-loop expansion in the alternative zero-momentum renormalization scheme defined in the massive disordered phase.In addition, we report numerical Monte Carlo simulations of a class of three-dimensional O(2)⊗O(2)-symmetric lattice models. The results provide further support to the existence of the O(2)⊗O(2) universality class predicted by the field-theoretical analyses.

We compute the Renormalization Group functions of a Landau-GinzburgWilson Hamiltonian with O(n)×O(m) symmetry up to five-loop in Minimal Subtraction scheme. The line n + (m, d), which limits the region of secondorder phase transition, is reconstructed in the framework of the ǫ = 4 − d expansion for generic values of m up to O(ǫ 5 ). For the physically interesting case of noncollinear but planar orderings (m = 2) we obtain n + (2, 3) = 6.1(6) by exploiting different resummation procedures. We substantiate this results re-analyzing six-loop fixed dimension series with pseudo-ǫ expansion, obtaining n + (2, 3) = 6.22(12). We also provide predictions for the critical exponents characterizing the second-order phase transition occurring for n > n + .

We study the crossover behaviors that can be observed in the high-temperature phase of three-dimensional dilute spin systems, using a field-theoretical approach. In particular, for randomly dilute Ising systems we consider the Gaussian-to-random and the pure-Ising-to-random crossover, determining the corresponding crossover functions for the magnetic susceptibility and the correlation length. Moreover, for the physically interesting cases of dilute Ising, XY, and Heisenberg systems, we estimate several universal ratios of scaling-correction amplitudes entering the high-temperature Wegner expansion of the magnetic susceptibility, of the correlation length, and of the zero-momentum quartic couplings.