$\phi^4$ lattice model with cubic symmetry in three dimensions: RG-flow and first order phase transitions

We study the $3$-component $\phi^4$ model on the simple cubic lattice in presence of a cubic perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite size scaling analysis of the data. The analysis of the renormalization group (RG)-flow of a dimensionless quantity provides us with the accurate estimate $Y_4 - \omega_2 =0.00081(7)$ for the difference of the RG-eigenvalue $Y_4$ at the $O(3)$-symmetric fixed point and the correction exponent $\omega_2$ at the cubic fixed point. We determine an effective exponent $\nu_{eff}$ of the correlation length that depends on the strength of the breaking of the $O(3)$ symmetry. Field theory predicts that depending on the sign of the cubic perturbation, the RG-flow is attracted by the cubic fixed point, or runs to an ever increasing amplitude, indicating a fluctuation induced first order phase transition. We demonstrate directly the first order nature of the phase transition for a sufficiently strong breaking of the $O(3)$ symmetry. We obtain accurate results for the latent heat, the correlation length in the disordered phase at the transition temperature and the interface tension for interfaces between one of the ordered phases and the disordered phase. We study how these quantities scale with the RG-flow, allowing quantitative predictions for weaker breaking of the $O(3)$ symmetry.