We study the nature of the multicritical point in the three-dimensional $\mathrm{O}\left(3\right)\oplus \mathrm{O}\left(2\right)$ symmetric Landau-Ginzburg-Wilson theory, which describes the competition of two order parameters that are O(3) and O(2) symmetric, respectively. This study is relevant for SO(5) theory of high-$T_c$ superconductors, which predicts the existence of a multicritical point in the temperature-doping phase diagram, where the antiferromagnetic and superconducting transition lines meet. We investigate whether $\mathrm{O}\left(3\right)\oplus \mathrm{O}\left(2\right)$ symmetry gets effectively enlarged to O(5) approaching the multicritical point. For this purpose, we study the stability of the O(5) fixed point. By means of a Monte Carlo simulation, we show that the O(5) fixed point is unstable with respect to the spin-4 quartic perturbation with the crossover exponent ${\varphi }_{4,4}=0.180\left(15\right)$, in substantial agreement with recent field-theoretical results. This estimate is much larger than the one-loop $ϵ$-expansion estimate ${\varphi }_{4,4}=1∕26$, which has often been used in the literature to discuss the multicritical behavior within SO(5) theory. Therefore, no symmetry enlargement is generically expected at the multicritical transition. We also perform a five-loop field-theoretical analysis of the renormalization-group flow. It shows that bicritical systems are not in the attraction domain of the stable decoupled fixed point. Thus, in these systems—high-$T_c$ cuprates should belong to this class—the multicritical point corresponds to a first-order transition.

• Received 17 February 2005

DOI:https://doi.org/10.1103/PhysRevB.72.014532