We study the multicritical behavior arising from the competition of two distinct types of ordering characterized by O(n) symmetries. For this purpose, we consider the renormalization-group flow for the most general O(n1)⊕O(n2)symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields φ1 and φ2 with n1 and n2 components respectively. In particular, we determine in which cases, approaching the multicritical point, one may observe the asymptotic enlargement of the symmetry to O(N) with N = n1 + n2. By performing a five-loop ǫ-expansion computation we determine the fixed points and their stability. It turns out that for N = n1 + n2 ≥ 3 the O(N)symmetric fixed point is unstable. For N = 3, the multicritical behavior is described by the biconal fixed point with critical exponents that are very close to the Heisenberg ones. For N ≥ 4 and any n1,n2 the critical behavior is controlled by the tetracritical decoupled fixed point. We discuss the relevance of these results for some physically interesting systems, in particular for anisotropic antiferromagnets in the presence of a magnetic field and for high-Tc superconductors. Concerning the SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point is unstable with a significant crossover exponent, φ4,4 ≈ 0.15; this implies that the O(5) symmetry is not effectively realized at the point where the antiferromagnetic and superconducting transition lines meet. The multicritical behavior is either governed by the tetracritical decoupled fixed point or is of first-order type if the system is outside its attraction domain
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