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

We show that phase transitions in spin-one Bose gases and stacked triangular Heisenberg antiferromagnets -- an example of frustrated magnets with competing interactions -- are described by the same Landau-Ginzburg-Wilson Hamiltonian with O(3)$times$O(2) symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional O(3)$times$O(2) model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-one Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent $ u$ is fully determined by the scattering lengths $a_0$ and $a_2$. We provide estimates of $ u$ in $^{87}$Rb, $^{41}$K and $^7$Li atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover 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 (with complex coordinates) which slow down the RG flow ($^7$Li). These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the O($N$)$times$O(2) model when $N$ is larger than $N_csimeq 5.3$ (the transition 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 Monte Carlo simulations yielding an apparent second-order transition.