We study the $2$-component $\phi^4$ model on the simple cubic lattice in the presence of a cubic, or equivalently, a $\mathbb{ D}_4$ invariant perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite size scaling analysis of the data. We follow previous work on the $3$-component case. We study the RG flow from the decoupled Ising fixed point into the $O(2)$-invariant one and towards the fluctuation induced first order transition. To this end we study the behavior of phenomenological couplings. At the $O(2)$-invariant fixed point we obtain the estimate $Y_4=-0.1118(10)$ of the RG-exponent of the perturbation. Note that the small modulus of $Y_4$ means that the RG flow is slow. Hence, in order to interpret experiments or Monte Carlo simulations of lattice models, which are effectively described by the $\phi^4$ model with a cubic term, we have to consider the RG flow beyond the neighborhood of the fixed points.
Categories: cond-mat.stat-mech hep-lat hep-th