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We show that phase transitions in spin-one Bose gases and stacked triangularHeisenberg antiferromagnets -- an example of frustrated magnets with competinginteractions -- are described by the same Landau-Ginzburg-Wilson Hamiltonianwith O(3)$\times$O(2) symmetry. In agreement with previousnonperturbative-renormalization-group studies of the three-dimensionalO(3)$\times$O(2) model, we find that the transition from the normal phase tothe superfluid ferromagnetic phase in a spin-one Bose gas is weakly first orderand shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent$\nu$ is fully determined by the scattering lengths $a_0$ and $a_2$. We provideestimates of $\nu$ in $^{87}$Rb, $^{41}$K and $^7$Li atom gases which can betested experimentally. We argue that pseudoscaling comes from either acrossover phenomena due to proximity of the O(6) Wilson-Fisher fixed point($^{87}$Rb and $^{41}$K) or the existence of two unphysical fixed points (withcomplex coordinates) which slow down the RG flow ($^7$Li). These unphysicalfixed points are a remnant of the chiral and antichiral fixed points that existin the O($N$)$\times$O(2) model when $N$ is larger than $N_c\simeq 5.3$ (thetransition being then second order and controlled by the chiral fixed point).Finally, we discuss a O(2)$\times$O(2) lattice model and show that our results,even though we find the transition to be first order, are compatible with MonteCarlo simulations yielding an apparent second-order transition.

First-order phase transitions in spinor Bose gases and frustrated magnets