Spontaneous symmetry breaking on the lattice generated by Yukawa interaction

We study by numerical simulation a lattice Yukawa model with naive fermions at intermediate values of the Yukawa coupling $y$ when the nearest neighbour coupling $\text{kp}$ of the scalar field $\mathrm{\Phi }$ is very weakly ferromagnetic ($\kp \approx 0$)orevenantiferromagnetic($\kappa < 0$) and the nonvanishing value of $\text{vev}$ is generated by the Yukawa interaction. The renormalized Yukawa coupling $y_R$ achieves here its maximal value and this $y$-region is thus of particular importance for lattice investigations of strong Yukawa interaction. However, here the scalar field propagators have a very complex structure caused by fermion loop corrections and by the proximity of phases with antiferromagnetic properties. We develop methods for analyzing these propagators and for extracting the physical observables. We find that going into the negative $\text{kp}$ region, the scalar field renormalization constant becomes small and $y_R$ does not seem to exceed the unitarity bound, making the existence of a nontrivial fixed point in the investigated Yukawa model quite unlikely.