Critical line of the ${\mathrm{\Phi }}^4$ theory on a simple cubic lattice in the local potential approximation

We establish the critical line of the one-component ${\mathrm{\Phi }}^4$ (or Landau-Ginzburg) model on the simple cubic lattice in three dimensions. Our study is performed in the framework of the non-perturbative renormalization group in the local potential approximation. Soft as well as ultra-sharp infra-red regulators are both considered. While the latter gives poor results, the critical line given by the soft cut-off compares well with the Monte Carlo simulations data of Hasenbusch (J. Phys. A : Math. Gen. 32 (1999) 4851) with a relative error of, at worst, $\sim {3.10}^{-3}$ on published points (critical parameters) of this line.