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We investigate a U (1) lattice chiral gauge theory (LχGT ) with domain wall fermions and compact gauge fixing. In the reduced model limit, our perturbative and numerical investigations show that there exist no extra mirror chiral modes. The longitudinal gauge degrees of freedom have no effect on the free domain wall fermion spectrum consisting of opposite chiral modes at the domain wall and at the anti-domain wall which have an exponentially damped overlap.

We study by numerical simulation a lattice Yukawa model with naive fermions at intermediate values of the Yukawa coupling constant y when the nearest neighbour coupling κ of the scalar field Φ is very weakly ferromagnetic (κ ≈ 0) or even antiferromagnetic (κ < 0) and the nonvanishing value of Φ 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 κ 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.