Cubic fixed point in three dimensions: Monte Carlo simulations of the $\phi^4$ model on the lattice

We study the cubic fixed point for $N=3$ and $4$ by using finite size scaling applied to data obtained from Monte Carlo simulations of the $N$-component $\phi^4$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the $O(N)$-invariance. For $N=4$, we determine the correction exponents $\omega_1=0.763(24)$ and $\omega_2=0.082(5)$. In the case of $N=3$, we obtain $Y_4=0.0142(6)$ for the RG-exponent of the cubic perturbation at the $O(3)$-invariant fixed point, while the correction exponent $\omega_2=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\nu=0.7202(7)$ and $\eta=0.0371(2)$ of the critical exponents of the cubic fixed point for $N=4$. For $N=3$, at the cubic fixed point, the $O(3)$-symmetry is only mildly broken and the critical exponents differ only by little from those of the $O(3)$-invariant fixed point. We find $-0.00001 \lessapprox \eta_{cubic}- \eta_{O(3)} \lessapprox 0.00007$ and $\nu_{cubic}-\nu_{O(3)} =-0.00061(10)$.